Multiplying Polynomials

Polynomial multiplication is a process for multiplying together two or more polynomials. We can perform polynomial multiplication by applying the distributive property to the multiplication of polynomials.

To multiply two polynomials with each other, take the terms of the first polynomial and distribute them over the second polynomial.

$$(a+b)(c+d) = a(c+d) + b(c+d) = ac+ad+bc+bd.$$

Alternatively, distribute the terms of the second polynomial:

$$(a+b)(c+d) = (a+b)c + (a+b)d = ac+bc+ad+bd.$$

Although the terms are in slightly different order, these two results are the same.

Below are some basic example of polynomial multiplication.

Expand $(x-3)(y+2)$.

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$$\begin{aligned} (x-3)(y+2) &= (x -3)y + (x-3) \times 2 \\ & = xy - 3y + 2x -6.\ _\square \end{aligned}$$

Let's see some slightly harder examples:

Expand $(a+b)(x+y+z)$.

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We have

$$\begin{aligned} (a+b)(x+y+z) &= (a+b)x + (a+b)y + (a+b)z \\ &= ax + bx + ay + by + az + bz. \end{aligned}$$

Alternatively, you can also solve by distributing $(a + b)$:

$$\begin{aligned} (a+b)(x+y+z) &= a(x + y + z) + b(x + y + z) \\ &= ax + ay + az + bx + by + bz. \end{aligned}$$

Although the terms are in slightly different order, these two results are the same. $_\square$

Expand $(2x+1)(x^{2}+3x+4).$

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Expand the list by the method shown above:

$$2x(x^{2}+3x+4)+1(x^{2}+3x+4)=2x^{3}+6x^{2}+8x+x^{2}+3x+4.$$

Collecting like terms, we have

$$2x^{3}+(6x^{2}+x^{2})+(8x+3x)+4=2x^{3}+7x^{2}+11x+4.\ _\square$$

The following forms come up a lot in algebra. We will go over how to expand them in the examples below, but you should also take some time to store these forms in memory, since you'll see them often.

1. $(a \pm b)^2 = a^2 \pm 2ab + b^2$
2. $(a+b)(a-b) = a^2 - b^2$
3. $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$
4. $(a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3$
5. $(a \pm b)(a^2 \mp ab + b^2) = a^3 \pm b^3$

Expand $(x+2)^2$.

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We have

$$\begin{aligned} (x+2)^2 &= (x+2)(x+2) \\ &= x(x+2) + 2(x+2) \\ & = x^2 + 2x + 2x +4 \\ &= x^2 + 4x +4. \end{aligned}$$

Expand $(a+b)^2$.

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We have

$$\begin{aligned} (a+b)^2 &= (a+b)(a+b) \\ &= a(a+b) + b(a+b) \\ & = a^2 + ab + ba +b^2 \\ &= a^2 + 2ab +b^2.\ _\square \end{aligned}$$

Expand $(-a-b)^2$.

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Since we know from the previous example that $(a+b)^2 = a^2 + 2ab + b^2$, we conclude that

$$(-a-b)^2 = (a+b)^2 = a^2 + 2ab +b^2.\ _\square$$

Expand $(a-b)^2$.

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We have

$$\begin{aligned} (a-b)^2 &= (a-b)(a-b) \\ &= a(a-b) - b(a-b) \\ & = a^2 - ab - ba +b^2 \\ &= a^2 - 2ab +b^2.\ _\square \end{aligned}$$

Expand $(-a+b)^2$.

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We have

$$\begin{aligned} (-a+b)^2 &= (-(a-b))^2 \\ &= ((-1^2) \times (a-b)^2) \\ & = (a-b)^2 . \end{aligned}$$

Since we know from the previous example that $(a-b)^2 = a^2 - 2ab + b^2$, we conclude that

$$(-a+b)^2 = (a-b)^2 = a^2 - 2ab +b^2.\ _\square$$

Expand $(a+b)(a-b)$.

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We have

$$\begin{aligned} (a+b)(a-b) &= a(a-b) + b(a-b) \\ &= a^2 - ab + ab - b^2 \\ & = a^2 - b^2.\ _\square \end{aligned}$$

Expand $(x+a)(x+b)$.

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We have

$$\begin{aligned} (x+a)(x+b) &= x(x+b) + a(x+b) \\ &= x^2 + bx + ax + ab \\ & = x^2 + (a+b)x + ab.\ _\square \end{aligned}$$

Expand $(x+2)(x+3)$.

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We have

$$\begin{aligned} (x+2)(x+3) &= x^2 + (2+3)x + (2 \times 3)\\ & = x^2 + 5x + 6.\ _\square \end{aligned}$$

Expand $(x-2)(x+3)$.

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We have

$$\begin{aligned} (x-2)(x+3) &= x^2 + (-2+3)x + (-2 \times 3)\\ & = x^2 + x - 6.\ _\square \end{aligned}$$

The leading coefficient of a polynomial is the coefficient on the variable with the highest power. Let's see an example where you must simplify to find the leading coefficient. This is a common type of problem in algebra.

What is the leading coefficient in the expansion of $(2x+5x^{2})(4x^{3}+2x^{2}+3x^{4})$?

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Expand the list:

$$\begin{aligned} &2x(4x^{3}+2x^{2}+3x^4)+5x^{2}(4x^{3}+2x^{2}+3x^4)\\ =&8x^{4}+4x^{3}+6x^{5}+20x^{5}+10x^{4}+15x^{6}. \end{aligned}$$

Collecting like terms, we have

$$\begin{aligned} &(8x^{4}+10x^{4})+4x^{3}+(6x^{5}+20x^{5})+15x^{6}\\ =&18x^{4} + 4x^{3}+ 26x^{5} + 15x^{6}. \end{aligned}$$

Since the term with the highest power is $x^{6},$ the leading coefficient is $15.\ _\square$

As the number of terms gets bigger, the work becomes messy so we must take care to not make any errors in signs and arithmetic.

What is the sum of all the coefficients in the expansion of

$$(3x ^{2} + 4x + 2)(x^{ 2} - 4x + 3)?$$

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Expand the list:

$$\begin{aligned} &3x^{2}(x^{ 2} - 4x + 3)+4x(x^{ 2} - 4x + 3)+2(x^{ 2} - 4x + 3)\\ =&3x^{4}-12x^{3}+9x^{2}+4x^{3}-16x^{2}+12x+2x^{2}-8x+6. \end{aligned}$$

Collecting like terms, we have

$$\begin{aligned} &3x^{4}+(-12x^{3}+4x^{3})+(9x^{2}-16x^{2}+2x^{2})+(12x-8x)+6\\ =&3x^{4}-8x^{3}-5x^{2}+4x+6. \end{aligned}$$

Therefore the sum is $3-8-5+4+6=0.\ _\square$

• Distributive Property
• Polynomial Division
• Polynomial Roots