Multiply each expression. 3 a ( 5 a + 3 b ) 3a(5a+3b)

15 a 2 + 9 a b 15a^2+9ab 15 a 2 + 9 ab

9 a 2 + 9 a b 9a^2+9ab 9 a 2 + 9 ab

Multiply each expression. ( 3 x 2 − 5 x + 3 ) 2 x 2 (3x^2-5x+3)2x^2

6 x 4 − 10 3 + 6 x 2 6x^4-10^3+6x^2

6 x 4 + 10 x 3 + 6 x 2 6x^4+10x^3+6x^2

6 x 3 − 10 x 2 + 6 x 6x^3-10x^2+6x

6 x 4 − 10 x 2 + 6 x 6x^4-10x^2+6x

Multiply the polynomials using FOIL. ( x − 5 ) ( x − 12 ) \left(x-5\right)\left(x-12\right)

x 2 − 5 x + 60 x^2-5x+60 x 2 − 5 x + 60

x 2 − 17 x − 60 x^2-17x-60 x 2 − 17 x − 60

Multiply the polynomials using FOIL. ( 4 x + 7 ) ( − x + 6 ) \left(4x+7\right)\left(-x+6\right)

− 4 x 2 + 17 x + 42 -4x^2+17x+42 − 4 x 2 + 17 x + 42

− 4 x 2 − x + 42 -4x^2-x+42 − 4 x 2 − x + 42

Multiply the polynomials using FOIL. ( x 2 − 3 x ) ( 2 x + 8 ) \left(x^2-3x\right)\left(2x+8\right)

2 x 3 + 2 x 2 − 6 x − 24 2x^3+2x^2-6x-24

Multiply the polynomials by using FOIL. ( 5 x y + 3 ) ( 4 x y − 2 ) (5xy+3)(4xy-2)

20 x 2 y 2 + 2 x y − 6 20x^2y^2+2xy-6

20 x 2 y 2 − 2 x y − 6 20x^2y^2-2xy-6

20 x 2 y 2 + 6 x y − 6 20x^2y^2+6xy-6

20 x 2 y 2 − 6 x y − 6 20x^2y^2-6xy-6 20 x 2 y 2 − 6 x y − 6

Multiply the polynomials. ( x + 4 ) ( 3 x 2 − 2 x + 1 ) \left(x+4\right)\left(3x^2-2x+1\right)

3 x 3 + 10 x 2 − 7 x + 4 3x^3+10x^2-7x+4 3 x 3 + 10 x 2 − 7 x + 4

3 x 3 + 12 x 2 + x + 4 3x^3+12x^2+x+4 3 x 3 + 12 x 2 + x + 4

12 x 2 − 8 x + 4 12x^2-8x+4 12 x 2 − 8 x + 4

3 x 3 − 2 x 2 + x 3x^3-2x^2+x 3 x 3 − 2 x 2 + x

Multiply the polynomials. ( x + 3 ) ( x − 5 ) ( − 2 x + 1 ) \left(x+3\right)\left(x-5\right)\left(-2x+1\right)

− 2 x 3 + 5 x 2 + 28 x − 15 -2x^3+5x^2+28x-15 − 2 x 3 + 5 x 2 + 28 x − 15

− 2 x 3 + 4 x 2 + 30 x -2x^3+4x^2+30x

2 x 3 + 5 x 2 + 28 x − 15 2x^3+5x^2+28x-15 2 x 3 + 5 x 2 + 28 x − 15

6. Exponents, Polynomials, and Polynomial Functions

Multiplying Polynomials

6. Exponents, Polynomials, and Polynomial Functions

Multiplying Polynomials: Videos & Practice Problems

Video Lessons

Topic summary

Multiplying polynomials involves using the distributive property to multiply each term, whether dealing with monomials or multivariable polynomials. The FOIL method efficiently multiplies two binomials by combining the First, Outer, Inner, and Last terms. For polynomials with more than two terms, distribute each term of the smaller polynomial across the larger one, then combine like terms. The number of resulting terms before simplification equals the product of the number of terms in each polynomial. Understanding coefficients, exponents, and terms is essential for simplifying expressions and mastering polynomial multiplication.

concept

Multiply Polynomials by Monomials

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Multiply Polynomials by Monomials Video Summary

Multiplying polynomials by monomials can be efficiently handled using the distributive property, which states that a term multiplied by a sum of terms can be distributed across each term in the sum. This approach simplifies the multiplication process by breaking it down into manageable parts.

For example, when multiplying a monomial like 4x by a binomial such as 3x - 7, apply the distributive property by multiplying 4x with each term inside the parentheses. This results in:

\[4x \times 3x - 4x \times 7 = 12x^2 - 28x\]

Here, the coefficients multiply directly (4 × 3 = 12), and the variables multiply by adding their exponents (x × x = x²). Similarly, the second term is simplified by multiplying the coefficient 4 by 7 and retaining the variable x.

When the polynomial has more terms, such as y² + 3y + 2, and is multiplied by a monomial like 5y², the distributive property still applies regardless of the order of multiplication. Multiply the monomial by each term of the polynomial:

\[5y^2 \times y^2 + 5y^2 \times 3y + 5y^2 \times 2 = 5y^4 + 15y^3 + 10y^2\]

In this case, multiply the coefficients (5 × 1, 5 × 3, 5 × 2) and add exponents of like bases (y² × y² = y⁴, y² × y = y³). This systematic use of the distributive property ensures accurate multiplication of polynomials by monomials, regardless of the number of terms or their position.

Understanding this method lays the foundation for more complex polynomial multiplication and reinforces the importance of properties like distribution in algebraic operations.

Problem

Multiply each expression.
$3 a (5 a + 3 b)$

$15 a squared plus 3 a b$

$9 a squared plus 15 a b$

$15a^2+9ab$

$9a^2+9ab$

Problem

Multiply each expression.
$- x (3 x^{2} - 4 x + 2)$

$-3x^3+4x^2-2x$

$3x^3-4x^2+2x$

$-3x^3-4x^2-2x$

$3x^3+4x^2+2x$

Problem

Multiply each expression.
$(3 x^{2} - 5 x + 3) 2 x^{2}$

$6 x^{4} + 10 x^{3} + 6 x^{2}$

$6 x^{3} - 10 x^{2} + 6 x$

$6 x^{4} - 10 x^{2} + 6 x$

$6 x^{4} - 10^{3} + 6 x^{2}$

concept

Multiply Binomials Using the FOIL Method

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Multiply Binomials Using the FOIL Method Video Summary

Multiplying polynomials, particularly binomials, can be efficiently accomplished using the FOIL method, which stands for First, Outer, Inner, and Last. This technique is essential for simplifying expressions that involve two-term polynomials.

To begin, identify the two binomials you want to multiply. For example, consider the binomials $ (x + 2) $ and $ (x + 3) $. The FOIL method guides you through the multiplication process:

First: Multiply the first terms of each binomial. In this case, $ x $ and $ x $ yield $ x^2 $.
Outer: Multiply the outer terms, which are $ x $ and $ 3 $, resulting in $ 3x $.
Inner: Multiply the inner terms, $ 2 $ and $ x $, giving $ 2x $.
Last: Finally, multiply the last terms, $ 2 $ and $ 3 $, which results in $ 6 $.

After applying the FOIL method, you combine all the results: $ x^2 + 3x + 2x + 6 $. The next step is to simplify the expression by combining like terms. Here, $ 3x $ and $ 2x $ combine to form $ 5x $. Thus, the final simplified expression is:

\[ x^2 + 5x + 6 \]

This process illustrates how to multiply and simplify binomials effectively. Mastering the FOIL method is crucial for tackling more complex polynomial multiplication in future mathematical endeavors.

Problem

Multiply the polynomials using FOIL.
$(x - 5) (x - 12)$

$x^{2} - 12 x + 60$

$x^{2} - 17 x + 60$

$x^2-5x+60$

$x^2-17x-60$

Problem

Multiply the polynomials using FOIL.
$(4 x + 7) (- x + 6)$

$4 x^{2} + 17 x + 42$

$4 x^{2} + 31 x + 42$

$-4x^2+17x+42$

$-4x^2-x+42$

Problem

Multiply the polynomials using FOIL.
$(x^{2} - 3 x) (2 x + 8)$

$2 x^{2} + 2 x - 24$

$2 x^{3} + 2 x^{2} - 6 x - 24$

$2 x^{3} - 2 x^{2} - 24 x$

$2 x^{3} + 2 x^{2} - 24 x$

example

Multiply Binomials Using the FOIL Method Example 1

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Multiply Binomials Using the FOIL Method Example 1 Video Summary

To multiply the binomials 2x + y and xy - 3, the FOIL method is applied, which stands for First, Outside, Inside, and Last terms multiplication. First, multiply the first terms of each binomial: 2x times xy, resulting in 2x²y. Next, multiply the outside terms: 2x times -3, giving -6x. Then, multiply the inside terms: y times xy, which simplifies to xy². Finally, multiply the last terms: y times -3, resulting in -3y.

Combining these results, the expanded expression is 2x²y - 6x + xy² - 3y. This process demonstrates how to expand binomials by systematically multiplying each pair of terms and then combining like factors, such as recognizing that x × x = x² and y × y = y². The FOIL technique is a foundational algebraic method for polynomial multiplication, enhancing understanding of distributive properties and term combination.

Problem

Multiply the polynomials by using FOIL.
$(5 x y + 3) (4 x y - 2)$

$20 x^{2} y^{2} - 2 x y - 6$

$20 x^{2} y^{2} + 6 x y - 6$

$20 x^{2} y^{2} + 2 x y - 6$

$20x^2y^2-6xy-6$

example

Multiply Binomials Using the FOIL Method Example 2

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concept

Multiply Polynomials Using the Distributive Property

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Multiply Polynomials Using the Distributive Property Video Summary

Multiplying polynomials can be approached in several ways, depending on the number of terms involved. One common method is the distributive property, which is essential when dealing with expressions that contain more than two terms. For instance, when multiplying a two-term polynomial by a three-term polynomial, the process involves breaking down the shorter expression into its individual terms and distributing each term across the entire longer expression.

To illustrate, consider the expression $ (x + 3)(x^2 + x - 2) $. Instead of using the FOIL method, which is limited to two-term expressions, we can distribute each term in $ x + 3 $ to every term in $ x^2 + x - 2 $. This means we first take $ x $ and multiply it by each term inside the parentheses:

$ x \cdot (x^2 + x - 2) = x^3 + x^2 - 2x $

Next, we distribute the $ 3 $:

$ 3 \cdot (x^2 + x - 2) = 3x^2 + 3x - 6 $

Now, we combine the results from both distributions:

$ x^3 + x^2 - 2x + 3x^2 + 3x - 6 $

Combining like terms yields:

$ x^3 + (1x^2 + 3x^2) + (-2x + 3x) - 6 = x^3 + 4x^2 + x - 6 $

This final expression represents the simplified result of the multiplication. A useful pattern to note is that the number of terms in the resulting polynomial can be predicted by multiplying the number of terms in each polynomial before simplification. For example, multiplying a two-term polynomial by a three-term polynomial results in six terms before any simplification occurs. This serves as a helpful check to ensure that all terms have been accounted for during the multiplication process.

Problem

Multiply the polynomials.
$(x + 4) (3 x^{2} - 2 x + 1)$

$3x^3+10x^2-7x+4$

$3x^3+12x^2+x+4$

$12x^2-8x+4$

$3x^3-2x^2+x$

example

Multiply Polynomials Using the Distributive Property Example 3

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Multiply Polynomials Using the Distributive Property Example 3 Video Summary

When multiplying multiple polynomials, such as a monomial and two binomials, it is effective to first multiply the binomials using the FOIL method. FOIL stands for First, Outer, Inner, Last, referring to the terms in each binomial that are multiplied together. For example, multiplying the binomials x + 3 and -5x + 7 involves calculating:

$x \times (-5x) + x \times 7 + 3 \times (-5x) + 3 \times 7$

Simplifying these terms yields:

$-5x^2 + 7x - 15x + 21$

Combining like terms, specifically the linear terms \$7x$ and $-15x\(, results in:

\)-5x^2 - 8x + 21\(

Next, to multiply this trinomial by the monomial \)2x$, apply the Distributive Property by multiplying $2x\( with each term in the trinomial:

\)2x \times (-5x^2) + 2x \times (-8x) + 2x \times 21\(

Calculating each product gives:

\)-10x^3 - 16x^2 + 42x$

This final expression represents the product of the original three polynomials. Understanding how to multiply polynomials by combining methods like FOIL and the Distributive Property is essential for simplifying polynomial expressions and solving algebraic problems efficiently.

Problem

Multiply the polynomials.
$(x + 3) (x - 5) (- 2 x + 1)$

$x^{2} - 2 x - 15$

$- 2 x^{3} + 4 x^{2} + 30 x$

$-2x^3+5x^2+28x-15$

$2x^3+5x^2+28x-15$

example

Multiply Polynomials Using the Distributive Property Example 4

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Here’s what students ask on this topic:

The distributive property is a fundamental algebraic principle that states: $a (b + c) = a b + a c$ . When multiplying polynomials, this property allows you to multiply each term inside the parentheses by the term outside. For example, if you multiply a monomial by a polynomial, you distribute the monomial to each term of the polynomial. This ensures every term is accounted for systematically. This method works regardless of the number of terms in the polynomial or whether the monomial is on the left or right side. It is the foundation for multiplying polynomials and helps organize the multiplication process efficiently.

The FOIL method is a technique specifically for multiplying two binomials, which are polynomials with two terms each. FOIL stands for First, Outer, Inner, Last, referring to the pairs of terms you multiply:
- First: multiply the first terms of each binomial.
- Outer: multiply the outer terms.
- Inner: multiply the inner terms.
- Last: multiply the last terms.
For example, multiplying $x + 2$ and $x + 3$ involves:
$x^{2} + 3x + 2x + 6$ . Then combine like terms to get $x^{2} + 5x + 6$ . FOIL is a straightforward way to ensure all terms are multiplied correctly in binomial multiplication.

No, the FOIL method only works for multiplying two binomials (two-term polynomials). When dealing with polynomials that have more than two terms, FOIL is insufficient because it does not account for all term combinations. Instead, you use the distributive property by breaking the smaller polynomial into individual terms and multiplying each term by the entire larger polynomial. For example, multiplying a binomial by a trinomial involves distributing each term of the binomial across all three terms of the trinomial, then combining like terms. This approach ensures every term is multiplied properly and no terms are missed.

After multiplying polynomials, you often end up with several terms. Simplifying involves combining like terms—terms that have the same variable raised to the same power. For example, if you have terms like $3x^2$ and $5x^2$ , you add their coefficients to get $8x^2$ . This process reduces the expression to its simplest form. Remember, the number of terms before simplification equals the product of the number of terms in each polynomial. Simplifying helps in making the expression easier to understand and use in further calculations.

A useful way to check your multiplication is by counting the number of terms before simplification. Multiply the number of terms in the first polynomial by the number of terms in the second polynomial. The product is the number of terms you should have before combining like terms. For example, multiplying a binomial (2 terms) by a trinomial (3 terms) should give you 6 terms before simplification. If your result has fewer or more terms, you may have missed or added extra terms. This check helps ensure accuracy in your multiplication process.

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Multiplying Polynomials: Videos & Practice Problems

Multiply Polynomials by Monomials

Multiply Polynomials by Monomials Video Summary

Multiply each expression.
$3 a (5 a + 3 b)$

Multiply each expression.
$- x (3 x^{2} - 4 x + 2)$

Multiply each expression.
$(3 x^{2} - 5 x + 3) 2 x^{2}$

Multiply Binomials Using the FOIL Method

Multiply Binomials Using the FOIL Method Video Summary

Multiply the polynomials using FOIL.
$(x - 5) (x - 12)$

Multiply the polynomials using FOIL.
$(4 x + 7) (- x + 6)$

Multiply the polynomials using FOIL.
$(x^{2} - 3 x) (2 x + 8)$

Multiply Binomials Using the FOIL Method Example 1

Multiply Binomials Using the FOIL Method Example 1 Video Summary

Multiply the polynomials by using FOIL.
$(5 x y + 3) (4 x y - 2)$

Multiply Binomials Using the FOIL Method Example 2

Multiply Polynomials Using the Distributive Property

Multiply Polynomials Using the Distributive Property Video Summary

Multiply the polynomials.
$(x + 4) (3 x^{2} - 2 x + 1)$

Multiply Polynomials Using the Distributive Property Example 3

Multiply Polynomials Using the Distributive Property Example 3 Video Summary

Multiply the polynomials.
$(x + 3) (x - 5) (- 2 x + 1)$

Multiply Polynomials Using the Distributive Property Example 4

Here’s what students ask on this topic:

What is the distributive property and how is it used to multiply polynomials?

How does the FOIL method work when multiplying two binomials?

Can the FOIL method be used to multiply polynomials with more than two terms?

How do you simplify the product of two polynomials after multiplication?

How can you check if you have multiplied polynomials correctly?

Your Intermediate Algebra tutors

Multiplying Polynomials: Videos & Practice Problems

Multiply Polynomials by Monomials

Multiply Polynomials by Monomials Video Summary

Multiply each expression.3a(5a+3b)3a(5a+3b)

Multiply each expression.−x(3x2−4x+2)-x(3x^2-4x+2)

Multiply each expression.(3x2−5x+3)2x2(3x^2-5x+3)2x^2

Multiply Binomials Using the FOIL Method

Multiply Binomials Using the FOIL Method Video Summary

Multiply the polynomials using FOIL.(x−5)(x−12)\left(x-5\right)\left(x-12\right)

Multiply the polynomials using FOIL.(4x+7)(−x+6)\left(4x+7\right)\left(-x+6\right)

Multiply the polynomials using FOIL.(x2−3x)(2x+8)\left(x^2-3x\right)\left(2x+8\right)

Multiply Binomials Using the FOIL Method Example 1

Multiply Binomials Using the FOIL Method Example 1 Video Summary

Multiply the polynomials by using FOIL.(5xy+3)(4xy−2)(5xy+3)(4xy-2)

Multiply Binomials Using the FOIL Method Example 2

Multiply Polynomials Using the Distributive Property

Multiply Polynomials Using the Distributive Property Video Summary

Multiply the polynomials.(x+4)(3x2−2x+1)\left(x+4\right)\left(3x^2-2x+1\right)

Multiply Polynomials Using the Distributive Property Example 3

Multiply Polynomials Using the Distributive Property Example 3 Video Summary

Multiply the polynomials.(x+3)(x−5)(−2x+1)\left(x+3\right)\left(x-5\right)\left(-2x+1\right)

Multiply Polynomials Using the Distributive Property Example 4

Here’s what students ask on this topic:

What is the distributive property and how is it used to multiply polynomials?

What is the distributive property and how is it used to multiply polynomials?

How does the FOIL method work when multiplying two binomials?

How does the FOIL method work when multiplying two binomials?

Can the FOIL method be used to multiply polynomials with more than two terms?

Can the FOIL method be used to multiply polynomials with more than two terms?

How do you simplify the product of two polynomials after multiplication?

How do you simplify the product of two polynomials after multiplication?

How can you check if you have multiplied polynomials correctly?

How can you check if you have multiplied polynomials correctly?

Your Intermediate Algebra tutors

Multiply each expression.
$3 a (5 a + 3 b)$

Multiply each expression.
$- x (3 x^{2} - 4 x + 2)$

Multiply each expression.
$(3 x^{2} - 5 x + 3) 2 x^{2}$

Multiply the polynomials using FOIL.
$(x - 5) (x - 12)$

Multiply the polynomials using FOIL.
$(4 x + 7) (- x + 6)$

Multiply the polynomials using FOIL.
$(x^{2} - 3 x) (2 x + 8)$

Multiply the polynomials by using FOIL.
$(5 x y + 3) (4 x y - 2)$

Multiply the polynomials.
$(x + 4) (3 x^{2} - 2 x + 1)$

Multiply the polynomials.
$(x + 3) (x - 5) (- 2 x + 1)$