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 + 15 a b 9a^2+15ab 9 a 2 + 15 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 x 3 + 6 x 2 6x^4-10x^3+6x^2 6 x 4 − 10 x 3 + 6 x 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 3 − 10 x 2 + 6 x

6 x 4 − 10 x 2 + 6 x 6x^4-10x^2+6x 6 x 4 − 10 x 2 + 6 x

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

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

x 2 − 12 x + 60 x^2-12x+60 x 2 − 12 x + 60

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 − x + 42 -4x^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 − 24 x 2x^3+2x^2-24x 2 x 3 + 2 x 2 − 24 x

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

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

x 2 − 2 x − 15 x^2-2x-15 x 2 − 2 x − 15

− 2 x 3 + 4 x 2 + 30 x -2x^3+4x^2+30x − 2 x 3 + 4 x 2 + 30 x

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 and Polynomials

Multiplying Polynomials

6. Exponents and Polynomials

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 product's number of 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 process by breaking down the multiplication 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 4 × 7 = 28), and the variables follow the laws of exponents, where x × x = x^2.

Similarly, when multiplying a polynomial with multiple terms by a monomial, such as y^2 + 3y + 2 multiplied by 5y^2, distribute the monomial across each term:

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

In this case, the coefficients multiply as usual, and the exponents add according to the rule a^m \times a^n = a^{m+n}, so y^2 \times y^2 = y^{4} and y^2 \times y = y^{3}.

Understanding how to multiply polynomials by monomials using the distributive property is fundamental in algebra. It allows for systematic expansion of expressions regardless of the number of terms or the position of the monomial. Mastery of this concept lays the groundwork for more complex polynomial operations and simplifies algebraic manipulation.

Problem

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

$15 a squared plus 3 a b$

$9a^2+15ab$

$15a^2+9ab$

$9a^2+9ab$

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

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

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

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

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

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.
$(x - 5) (x - 12)$

$x^2-12x+60$

$x^2-17x+60$

$x^2-5x+60$

$x^2-17x-60$

Problem

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

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

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

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

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

Problem

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

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

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

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

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

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, Last. This technique helps in systematically multiplying each term in the first binomial by each term in the second binomial.

First, multiply the First terms: 2x and xy, resulting in 2x × xy = 2x²y. Next, multiply the Outside terms: 2x and -3, giving 2x × (-3) = -6x. Then, multiply the Inside terms: y and xy, which yields y × xy = xy². Finally, multiply the Last terms: y and -3, resulting in y × (-3) = -3y.

Combining these products, the expanded expression becomes:

\[2x^{2}y - 6x + xy^{2} - 3y\]

This expression represents the product of the two binomials fully expanded. Understanding how to apply the FOIL method is essential for multiplying binomials efficiently and accurately, especially when variables and coefficients are involved. Recognizing common factors, such as combining x × x into x² and y × y into y², helps simplify the expression and prepares it for further algebraic manipulation.

Problem

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

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

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

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

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

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 2

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Multiply Polynomials Using the Distributive Property Example 2 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. For example, multiplying the binomials x + 3 and -5x + 7 involves multiplying each term in the first binomial by each term in the second binomial: $x \times (-5x) + x \times 7 + 3 \times (-5x) + 3 \times 7$. Simplifying these products results in $-5x^2 + 7x - 15x + 21$. Combining like terms, the expression becomes $-5x^2 - 8x + 21$.

Next, to multiply this resulting polynomial by the monomial \$2x$, apply the Distributive Property by multiplying $2x$ by each term in the polynomial: $2x \times (-5x^2) + 2x \times (-8x) + 2x \times 21$. Simplifying each term yields $-10x^3 - 16x^2 + 42x$. This final expression represents the product of the original three polynomials.

This process highlights the importance of systematically applying multiplication rules for polynomials, including the FOIL method for binomials and the Distributive Property for monomial-polynomial multiplication. Understanding how to combine like terms and manage exponents is essential for simplifying polynomial expressions effectively.

Problem

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

$x^2-2x-15$

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

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

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

Here’s what students ask on this topic:

To multiply a polynomial by a monomial, use the distributive property. This means you multiply the monomial by each term inside the polynomial separately. For example, if you have a monomial $4x$ and a polynomial $3x - 7$ , multiply $4x$ by $3x$ and then by $- 7$ . This gives $12 x^{2} - 28 x$ . The key is to multiply coefficients and add exponents of like bases. This method works regardless of the number of terms in the polynomial or whether the monomial is on the left or right side.

The FOIL method is a technique to multiply two binomials, which are polynomials with two terms each. FOIL stands for First, Outer, Inner, Last, referring to the pairs of terms you multiply. For example, to multiply $x + 2$ and $x + 3$ , you multiply the First terms ( $x imes x = x^2$ ), Outer terms ( $x imes 3 = 3x$ ), Inner terms ( $2 imes x = 2x$ ), and Last terms ( $2 imes 3 = 6$ ). Then, combine like terms: $x^{2} + 3x + 2x + 6 = x^2 + 5x + 6$ . This method ensures all terms are multiplied systematically.

No, the FOIL method only works for multiplying two binomials (two-term polynomials). When multiplying polynomials with more than two terms, FOIL is insufficient because it does not cover all term combinations. Instead, you use the distributive property by breaking the smaller polynomial into individual terms and multiplying each term by the entire other polynomial. For example, multiplying a binomial by a trinomial involves distributing each term of the binomial across all terms of the trinomial, then combining like terms. This approach ensures every term is accounted for and the product is correct.

To multiply polynomials with different numbers of terms, distribute each term of the smaller polynomial across every term of the larger polynomial. For example, if you multiply a binomial $x + 3$ by a trinomial $x^2 + x - 2$ , multiply $x$ by each term of the trinomial, then multiply $3$ by each term. This results in multiple terms that you then combine by adding coefficients of like terms. The total number of terms before simplification equals the product of the number of terms in each polynomial (e.g., 2 terms × 3 terms = 6 terms). Simplifying by combining like terms gives the final polynomial product.

The number of terms in the product before simplification equals the product of the number of terms in each polynomial. For example, multiplying a polynomial with 2 terms by one with 3 terms results in 6 terms before combining like terms. This is because each term in the first polynomial multiplies every term in the second polynomial. This relationship helps verify that you have multiplied all terms correctly. If the number of terms in your product is less or more than this, you may have missed or added extra terms.

Your Beginning Algebra tutors

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 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 2

Multiply Polynomials Using the Distributive Property Example 2 Video Summary

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

Here’s what students ask on this topic:

How do you multiply a polynomial by a monomial using the distributive property?

What is the FOIL method and how is it used to multiply two binomials?

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

How do you multiply polynomials with different numbers of terms and simplify the result?

What is the relationship between the number of terms in the polynomials being multiplied and the number of terms in the product before simplification?

Your Beginning 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 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 2

Multiply Polynomials Using the Distributive Property Example 2 Video Summary

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

Here’s what students ask on this topic:

How do you multiply a polynomial by a monomial using the distributive property?

How do you multiply a polynomial by a monomial using the distributive property?

What is the FOIL method and how is it used to multiply two binomials?

What is the FOIL method and how is it used to multiply 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 multiply polynomials with different numbers of terms and simplify the result?

How do you multiply polynomials with different numbers of terms and simplify the result?

What is the relationship between the number of terms in the polynomials being multiplied and the number of terms in the product before simplification?

What is the relationship between the number of terms in the polynomials being multiplied and the number of terms in the product before simplification?

Your Beginning 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.
$(x + 3) (x - 5) (- 2 x + 1)$