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Section 8.1 Polynomial Functions

Subsection 1. Compute sums and products

Compare the rules for simplilfying products to the rules for simplifying sums.

Subsubsection Examples

Example 8.1.

Simplify each expression if possible.

  1. \(\displaystyle 3x^2-5x^3\)
  2. \(\displaystyle 3x^2(-5x^3)\)
Solution
  1. This expression is a difference of terms, but they are not like terms (because the variable has different exponents), so we cannot combine them.
  2. This expression is a product, and the powers have the same base, so we can apply the first law of exponents to get \(~3x^2(-5x^3)=-15x^5\text{.}\)
Example 8.2.

Simplify each expression if possible.

  1. \(\displaystyle -6t^4-8t^4\)
  2. \(\displaystyle -6t^4(-8t^4)\)
Solution
  1. This expression is a difference of like terms, so we can combine their coefficients to get \(~-6t^4-8t^4=-14t^4\)
  2. This expression is a product, and the powers have the same base, so we can apply the first law of exponents to get \(~-6t^4(-8t^4)=48t^8\)

Subsubsection Exercises

Simplify each expression if possible.

  1. \(\displaystyle 2a^2-9a^3+a^2\)
  2. \(\displaystyle 2a^2(-9a^3+a^2)\)
Answer
  1. \(\displaystyle 3a^2-9a^3\)
  2. \(\displaystyle -18a^5+2a^4\)

Simplify each expression if possible.

  1. \(\displaystyle 7-4a^3+2a^3\)
  2. \(\displaystyle 7-4a^2(2a^3)\)
Answer
  1. \(\displaystyle 7-2a^3\)
  2. \(\displaystyle 7-8a^5\)

Subsection 2. Use formulas

There are several useful formulas for simplifying polynomials.

Subsubsection Examples

Example 8.5.

If \(~a=5t^4,~\) find \(a^3\) and \(3a^2\text{.}\)

Solution

We substitute \(5t^4\) for \(a\) to find

\begin{align*} a^3 \amp =(\blert{5t^4})^3 = 5^3(t^4)^3 = 125t^{12} \amp \amp \blert{\text{Apply the third law of exponents.}}\\ 3a^2 \amp = 3(\blert{5t^4})^2 = 3 \cdot 5^2(t^4)^2 = 75t^8 \end{align*}
Example 8.6.

If \(a=2y\) and \(b=-3z^2\text{,}\) find \(b^3\) and \(3a^2b\text{.}\)

Solution

We substitute \(2y\) for \(a\) and \(-3z^2\) for \(b\) to find

\begin{align*} b^3 \amp =(\blert{-3z^2})^3 = (-3)^3(z^2)^3 = -27z^6 \\ 3a^2b \amp = 3 (\blert{2y})^2(\blert{-3z^2}) = 3(4y^2)(-3z^2)=-36y^2z^2 \end{align*}

Subsubsection Exercises

If \(~a=-4x^3~\) and \(~b=3h\text{,}\) find \(a^3\) and \(ab^2\text{.}\)

Answer
\(-64x^9;~~-36x^3h^2\)

If \(x=6p^2\) and \(y=mq^2\text{,}\) find \(y^3\) and \(x^2y\text{.}\)

Answer
\(m^3q^6;~~-36mp^4q^2\)

Subsection 3. Square binomials

Sometimes it is easier to use formulas to square binomials.

Special Products of Binomials.
\begin{align*} \amp(a + b)^2 = (a + b) (a + b) = a^2 + 2ab + b^2\\ \amp(a - b)^2 = (a - b) (a - b) = a^2 - 2ab + b^2\\ \amp(a + b) (a - b)= a^2 -b^2 \end{align*}

Subsubsection Examples

Example 8.9.

Use the identity \(~(a+b)^2=a^2+2ab+b^2~\) to expand \(~(3h^2+4k^3)^2\text{.}\)

Solution

We substitute \(3h^2\) for \(a\) and \(4k^3\) for \(b\) into the identity.

\begin{align*} (3h^2+4k^3)^2 \amp =(3h^2)^2 +2(3h^2)(4k^3) +(4k^3)^2\\ \amp = 9h^4+24h^2k^3+16k^6 \end{align*}
Example 8.10.

Use the identity \(~(a-b)^2=a^2-2ab+b^2~\) to expand \(~(2xy^2-5)^2\text{.}\)

Solution

We substitute \(2xy^2\) for \(a\) and \(5\) for \(b\) into the identity.

\begin{align*} (2xy^2-5)^2 \amp =(2xy^2)^2 -2(2xy^2)(5) +5^2\\ \amp = 4x^2y^4 - 20xy^2+25 \end{align*}

Subsubsection Exercises

Expand \(~(8w^4-3w^3)^2\)

Answer
\(64w^8-48w^7+9w^6\)

Expand \(~(a^3b+9ab^3)^2\)

Answer
\(a^6b^2+18a^4b^4+81a^2b^6\)