Multiplying Polynomials: Videos & Practice Problems

Multiplying polynomials involves methods like the distributive property and the FOIL method for binomials. The FOIL acronym stands for First, Outer, Inner, and Last, guiding the multiplication of two binomials. For polynomials with more than two terms, distribute each term of the shorter polynomial across the longer one. Special product formulas, such as the difference of squares and squares of binomials, simplify multiplication by providing quick solutions. Recognizing these patterns can save time and effort in calculations.

FOIL

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 apply the FOIL method, start by identifying the two binomials you want to multiply. For example, consider the binomials \( (x + 2) \) and \( (x + 3) \). The process involves four steps:

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

After performing these multiplications, you combine 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 method not only streamlines the multiplication of binomials but also reinforces the importance of combining like terms to achieve a simplified polynomial. Understanding and mastering the FOIL method is crucial for further studies in algebra and polynomial functions.

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

• A. $x^2-12x+60$
• B. $x^2-17x+60$
• C. $x^2-5x+60$
• D. $x^2-17x-60$

Answer: (B)

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

• A. $4x^2+17x+42$
• B. $4x^2+31x+42$
• C. $-4x^2+17x+42$
• D. $-4x^2-x+42$

Answer: (C)

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

• A. $2x^2+2x-24$
• B. $2x^3+2x^2-6x-24$
• C. $2x^3-2x^2-24x$
• D. $2x^3+2x^2-24x$

Answer: (D)

Multiply Polynomials Using the Distributive Property

Multiplying polynomials can be approached in several ways, depending on the number of terms involved. One fundamental method is the distributive property, which applies when multiplying a single term by a polynomial. This involves distributing the term across all components of the polynomial inside the parentheses. For example, if you have a term like \(x\) multiplying a polynomial \(x^2 + x - 2\), you would multiply \(x\) by each term inside, resulting in \(x^3 + x^2 - 2x\).

Another common technique is the FOIL method, which stands for First, Outer, Inner, Last. This method is specifically designed for multiplying two binomials (two-term expressions). However, it is important to note that FOIL is not applicable for polynomials with more than two terms. Attempting to use FOIL on a polynomial like \( (x + 3)(x^2 + x - 2) \) would lead to incomplete multiplication, as not all terms would be accounted for.

When faced with a polynomial multiplication involving more than two terms, the best approach is to break down the shorter polynomial into its individual terms and apply the distributive property to each. For instance, with \( (x + 3)(x^2 + x - 2) \), you would treat it as two separate distributive problems: first distributing \(x\) across \(x^2 + x - 2\), and then distributing \(3\) across the same polynomial. This results in \(x^3 + x^2 - 2x\) from the first distribution and \(3x^2 + 3x - 6\) from the second.

After performing the distributions, the next step is to combine like terms. In this case, you would combine \(x^2\) terms and \(x\) terms to simplify the expression. The final result would be \(x^3 + 4x^2 + x - 6\).

A useful pattern to observe is the relationship between the number of terms in the original polynomials and the resulting terms after multiplication. For example, multiplying a polynomial with \(m\) terms by another with \(n\) terms will yield \(m \times n\) terms before simplification. This serves as a quick check to ensure that all terms have been accounted for during the multiplication process.

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

• A. $3x^3+10x^2-7x+4$
• B. $3x^3+12x^2+x+4$
• C. $12x^2-8x+4$
• D. $3x^3-2x^2+x$

Answer: (A)

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

• A. $x^2-2x-15$
• B. $-2x^3+4x^2+30x$
• C. $-2x^3+5x^2+28x-15$
• D. $2x^3+5x^2+28x-15$

Answer: (C)

Special Products - Square Formulas

Multiplying polynomials can often be simplified by recognizing special patterns, allowing for quicker calculations using special product formulas. One key pattern is the difference of squares, which applies when you have an expression in the form of \( (a + b)(a - b) \). The result of this multiplication is given by the formula:

\[ a^2 - b^2 \]

For example, if you multiply \( (x + 5)(x - 5) \), you can identify \( a \) as \( x \) and \( b \) as \( 5 \). Applying the formula yields:

\[ x^2 - 5^2 = x^2 - 25 \]

This method is much faster than the traditional foiling method, where you would multiply each term individually and then combine like terms.

Another important pattern is the squares of binomials, which applies to expressions like \( (a + b)^2 \) or \( (a - b)^2 \). The formulas for these are:

\[ (a + b)^2 = a^2 + 2ab + b^2 \]

\[ (a - b)^2 = a^2 - 2ab + b^2 \]

For instance, when calculating \( (x + 6)^2 \), you can identify \( a \) as \( x \) and \( b \) as \( 6 \). Using the first formula, you find:

\[ x^2 + 2(x)(6) + 6^2 = x^2 + 12x + 36 \]

Recognizing these patterns not only streamlines the multiplication process but also enhances your ability to solve polynomial equations efficiently. By practicing these special product formulas, you can improve your algebraic skills and tackle more complex problems with ease.

Multiply the polynomials using special product formulas.
$\left(5x-9\right)\left(5x+9\right)$

• A. $5x^2-81$
• B. $25x^2-81$
• C. $25x^2+90x+81$
• D. $25x^2-90x+81$

Answer: (B)

Multiply the polynomials using special product formulas.
$\left(3x+5\right)\left(3x+5\right)$

• A. $9x^2+15x+25$
• B. $9x^2+25$
• C. $9x^2+30x+25$
• D. $9x^2+55$

Answer: (C)

Special Products - Cube Formulas

In algebra, understanding the special product formulas for cubes of binomials is essential for simplifying expressions efficiently. The formulas for the cubes of binomials are expressed as follows:

For the sum of cubes: (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

For the difference of cubes: (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

When applying these formulas, the signs in the expansion differ based on whether the binomial is a sum or a difference. For a sum, all terms are positive, while for a difference, the signs alternate starting with a negative.

For example, consider the expression (x - 3)^3 . Here, we identify a = x and b = 3 . Using the difference of cubes formula, we can expand it as follows:

(x - 3)^3 = x^3 - 3(x^2)(3) + 3(x)(3^2) - 3^3

Calculating each term gives:

• x^3
• - 9x^2 (since 3 \cdot 3 = 9 )
• 27x (since 3 \cdot 9 = 27 )
• - 27 (since 3^3 = 27 )

Thus, the complete expansion results in:

x^3 - 9x^2 + 27x - 27

To remember the coefficients in these expansions, note that they follow the pattern of 1, 3, 3, 1 . Additionally, the powers of a decrease from 3 to 0, while the powers of b increase from 0 to 3. This systematic approach allows for quick and accurate expansions without the need for lengthy multiplication.

Multiply the polynomials using special product formulas.
$\left(2x+4\right)^3$

• A. $8x^3+16x^2+32x+64$
• B. $8x^3+24x^2+24x+64$
• C. $8x^3+48x^2+96x+64$
• D. $x^3+24x^2+96x+64$

Answer: (C)

Here’s what students ask on this topic:

What is the FOIL method in multiplying polynomials?

The FOIL method is a technique used to multiply two binomials. FOIL stands for First, Outer, Inner, and Last, which refers to the order in which you multiply the terms of the binomials. For example, to multiply (x + 2)(x + 3), you would:

1. Multiply the First terms: x * x = x²

2. Multiply the Outer terms: x * 3 = 3x

3. Multiply the Inner terms: 2 * x = 2x

4. Multiply the Last terms: 2 * 3 = 6

Then, combine like terms to get the final result: x² + 5x + 6.

How do you multiply polynomials with more than two terms?

To multiply polynomials with more than two terms, use the distributive property. Break the shorter polynomial into individual terms and distribute each term across the longer polynomial. For example, to multiply (x + 3)(x² + x - 2), distribute x and 3 separately:

x(x² + x - 2) = x³ + x² - 2x

3(x² + x - 2) = 3x² + 3x - 6

Combine like terms to get the final result: x³ + 4x² + x - 6.

What are special product formulas in polynomial multiplication?

Special product formulas simplify polynomial multiplication by recognizing patterns. Two common formulas are:

1. Difference of squares: (a + b)(a - b) = a² - b²

2. Square of a binomial: (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b²

These formulas allow you to quickly find the product without performing all the individual multiplications.

How do you multiply binomials using the distributive property?

To multiply binomials using the distributive property, distribute each term of the first binomial to each term of the second binomial. For example, to multiply (x + 2)(x + 3), distribute x and 2 separately:

x(x + 3) = x² + 3x

2(x + 3) = 2x + 6

Combine like terms to get the final result: x² + 5x + 6.

What is the difference of squares formula?

The difference of squares formula is a special product formula used to multiply binomials of the form (a + b)(a - b). The formula is:

(a + b)(a - b) = a² - b²

This formula simplifies the multiplication process by recognizing that the middle terms cancel out, leaving only the difference of the squares of the two terms.