8.2: Polynomial Multiplication

Polynomials can be classified as:

• Monomials if they contain one term.
• Binomials if they contain two term.
• Trinomials if they contain three term.
• Polynomials if they contain three or more terms.

There are no examples or homework in this section.

Multiplication of Two Monomials

Definition: Multiplying Two Monomials

To multiply two monomials, multiply the terms together by adding the exponents and multiplying the numeric coefficients.

Example 8.2.1

Multiply two monomials:

1. \((3x^2 )(6x^3 )\)
2. \((4x)(x)\)
3. \((−2x^3 )(−7x^4 )\)

Solution

1. \(\begin{array} &&(3x^2 )(6x^3 ) &\;\;\;\;\;\;\;\;\;\;\;\;\text{Example problem} \\ &(3)(6)(x^{2+3}) &\;\;\;\;\;\;\;\;\;\;\;\;\text{Multiply the coefficients and add the exponents on the variables using the Product Rule for Exponents} \\ &18x^5 &\;\;\;\;\;\;\;\;\;\;\;\;\text{Solution} \end{array}\)

2. \(\begin{array} &&(4x)(x) &\;\;\;\;\;\;\;\;\;\;\;\;\text{Example problem} \\ &(4)(1)(x^{1+1}) &\;\;\;\;\;\;\;\;\;\;\;\;\text{Multiply the coefficients and add the exponents. The coefficient on \(x\) is \(1\), and the exponent on each \(x\) is \(1\).} \\ &4x^2 &\;\;\;\;\;\;\;\;\;\;\;\;\text{Solution} \end{array}\)

3. \(\begin{array} &&(−2x^3 )(−7x^4 ) &\;\;\;\;\;\;\;\;\text{Example problem} \\ &(−2)(−7)(x^{3+4}) &\;\;\;\;\;\;\;\;\text{Multiply the coefficients and add the exponents.} \\ &14x^7 &\;\;\;\;\;\;\;\;\text{Solution} \end{array}\)

Exercise 8.2.1

Multiply two monomials:

1. \((−3x^4 )(9x^7 )\)
2. \((2x)(2x)\)
3. \((−4x^7 )(5x^5 )\)
4. \((−6x^2 )(−x^2 )\)

Multiplication of a Polynomial by a Monomial

Definition: Multiplying a Polynomial By a Monomial

To multiply a polynomial by a monomial, multiply all terms of the polynomial by the monomial. Keep any subtractions in the original polynomial with the term following the subtraction as the sign of the coefficient of the term.

Example 8.2.2

Multiply a polynomial by a monomial:

1. \(3x^2 (15x^2 − 5x)\)
2. \(−7x(3x^2 − 2x + 9)\)
3. \(5x(4x^3 − 2x^2 + x − 3)\)

Solution

1. \(\begin{array} &&3x^2 (15x^2 − 5x) &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Example problem} \\ &(3x^2 )(15x^2 ) + (3x^2 )(−5x) &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Multiply all terms of the polynomial by the monomial. Then simplify by multiplying the pairs of monomials.} \\ &45x^4 + (−15x^3 ) &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Simplify} \\ &45x^4 − 15x^3 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Solution} \end{array}\)

2. \(\begin{array} &&−7x(3x^2 − 2x + 9) &\;\;\;\;\;\;\;\;\;\;\text{Example problem} \\ &(−7x)(3x^2 ) + (−7x)(−2x) + (−7x)(9) &\;\;\;\;\;\;\;\;\;\;\text{Multiply the coefficients and add the exponents. The coefficient on \(x\) is \(1\), and the exponent on each \(x\) is \(1\).} \\ &−21x^3 + 14x^2 − 63x &\;\;\;\;\;\;\;\;\;\;\text{Solution} \end{array}\)

3. \(\begin{array} &&5x(4x^3 − 2x^2 + x − 3) &\text{Example problem} \\ &(5x)(4x^3 ) + (5x)(−2x^2 ) + (5x)(x) + (5x)(−3) &\text{Multiply the coefficients and add the exponents.} \\ &20x^4 − 10x^3 + 5x^2 − 15x &\text{Solution} \end{array}\)

Exercise 8.2.2

Multiply a polynomial by a monomial:

1. \((−6x)(x^2 − 3)\)
2. \((3x^4 )(2x^2 − x − 5)\)
3. \((−4x^5 )(x^4 − 3x^3 + 3x^2 − x − 7)\)
4. \((x^2 )(−x^3 − 12)\)

Multiplication of Two Binomials

Definition: Multiplying Two Binomials

To multiply two binomials, use the FOIL technique to multiply: first terms, outer terms, inner terms and the last terms. FOIL ensures that all terms in the first binomial are multiplied with all terms in the second binomial. The order of multiplication of terms does not matter since multiplication is commutative. Take care to combine any like terms to fully simplify the solution.

Example 8.2.3

Multiply two binomials:

1. \((3x − 4)(2x + 5)\)
2. \((5x^2 − 2)(5x^2 + 2)\)
3. \((7x^3 − 4x^2 )(x − 5)\)

Solution

1. \(\begin{array} &&(3x − 4)(2x + 5) &\;\;\;\;\;\;\;\;\;\;\text{Example problem} \\ &(3x)(2x) + (3x)(5) + (−4)(2x) + (−4)(5) &\;\;\;\;\;\;\;\;\;\;\text{FOIL the terms to multiply all terms in the first binomial by all terms in the second binomial.} \\ &6x^2 + 15x + (−8x) + (−20) &\;\;\;\;\;\;\;\;\;\;\text{Combine like terms and simplify} \\ &6x^2 + 7x − 20 &\;\;\;\;\;\;\;\;\;\;\text{Solution} \end{array}\)

2. \(\begin{array} &&(5x^2 − 2)(5x^2 + 2) &\;\;\;\;\text{Example problem} \\ &(5x^2 )(5x^2 ) + (5x^2 )(2) + (−2)(5x^2 ) + (−2)(2) &\;\;\;\;\text{FOIL the terms to multiply all terms in the first binomial by all terms in the second binomial.} \\ &25x^4 + 10x^2 + (−10x^2 ) + (−4) &\;\;\;\;\text{Combine like terms and simplify} \\ &25x^4 − 4 &\;\;\;\;\text{Solution} \end{array}\)

3. \(\begin{array} &&(7x^3 − 4x^2 )(x − 5) &\text{Example problem} \\ &(7x^3 )(x) + (7x^3 )(−5) + (−4x^2 )(x) + (−4x^2 )(−5) &\text{FOIL the terms to multiply all terms in the first binomial by all terms in the second binomial.} \\ &7x^4 + (−35x^3 ) + (−4x^3 ) + 20x^2 &\text{Combine like terms and simplify} \\ &7x^4 − 39x^3 + 20x^2 &\text{Solution} \end{array}\)

Exercise 8.2.3

Multiply two binomials:

1. \((2x − 3)(6x + 5)\)
2. \((3x^2 − 4)(3x^2 + 4)\)
3. \((−4x^5 − 2)(7x^3 + 3)\)
4. \((2x − 7)(3x − 8)\)

Multiplication of Two Polynomials

Definition: Multiplying Two Polynomials

To multiply two polynomials, use the distributive property to multiply every term in the first polynomial by every term in the second polynomial. Like terms are then combined to simplify the solution.

Example 8.2.4

Multiply two polynomials:

1. \((2x + 5)(3x^2 − 6x + 9)\)
2. \((2x^2 + 4x − 5)(3x − 2)\)
3. \((x^2 − x + 3)(2x^2 + 6x − 1)\)

Solution

1. \(\begin{array} &&(2x + 5)(3x^2 − 6x + 9) &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Example problem} \\ &(2x)(3x^2 ) + (2x)(−6x) + (2x)(9) + (5)(3x^2 ) + (5)(−6x) + (5)(9) &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{FOIL the terms to multiply all terms in the} \\ & &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \text{first binomial by all terms in the second binomial.} \\ &6x^3 + (−12x^2 ) + 18x + 15x^2 + (−30x) + 45 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Combine like terms and simplify} \\ &6x^3 + 3x^2 − 12x + 45 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Solution} \end{array}\)

2. \(\begin{array} &&(2x^2 + 4x − 5)(3x − 2) &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Example problem} \\ &(2x^2 )(3x) + (2x^2 )(−2) + (4x)(3x) + (4x)(−2) + (−5)(3x) + (−5)(−2) &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{FOIL the terms to multiply all terms in the} \\ & &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \text{first binomial by all terms in the second binomial.} \\ &6x^3 + (−4x^2 ) + 12x^2 + (−8x) + (−15x) + 10 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Combine like terms and simplify} \\ &6x^3 + 8x^2 − 23x + 10 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Solution} \end{array}\)

3. \(\begin{array} &&(x^2 − x + 3)(2x^2 + 6x − 1) &\text{Example problem} \\ &(x^2 )(2x^2 ) + (x^2 )(6x) + (x^2 )(−1) + (−x)(2x^2 ) + (−x)(6x) + (−x)(−1) + (3)(2x^2 ) + (3)(6x) + (3)(−1) &\text{FOIL the terms to multiply all terms in the} \\ & & \text{first binomial by all terms in the second binomial.} \\ &2x^4 + 6x^3 + (−1x^2 ) + (−2x^3 ) + (−6x^2 ) + x + 6x^2 + 18x + (−3) &\text{Combine like terms and simplify} \\ &2x^4 + 4x^3 − x^2 + 19x − 3 &\text{Solution} \end{array}\)

Exercise 8.2.4

Multiply two polynomials:

1. \((x^2 − 2x − 1)(2x^2 − 7x − 8)\)
2. \((3x^2 − 5)(x^2 + 4x − 3)\)
3. \((4x^3 − 2x + 1)(6x^2 + 3)\)
4. \((2x^3 − 3x + 4)(2x^2 − 8x + 2)\)