Add, Subtract, and Multiply Polynomial Expressions
When adding and subtracting polynomial expressions, you must first identify like terms which can then be combined. Like terms have the same base raised to the same exponent. An example of this would be \(5x^4\) and \(-8x^4\). In order to combine like terms, you will add or subtract the coefficients. It is often most helpful to write our answer in standard form (this was discussed in the last target).
Example 1: Simplify \(\left(-4x^3+7x-1\right)+\left(6x^3-5x^2+4x\right)\).
When adding and subtracting polynomial expressions, you must first identify like terms which can then be combined. Like terms have the same base raised to the same exponent. An example of this would be \(5x^4\) and \(-8x^4\). In order to combine like terms, you will add or subtract the coefficients. It is often most helpful to write our answer in standard form (this was discussed in the last target).
Example 1: Simplify \(\left(-4x^3+7x-1\right)+\left(6x^3-5x^2+4x\right)\).
\(\begin{align}&\left(-4x^3+7x-1\right)+\left(6x^3-5x^2+4x\right)\ & \ \ &\text{1) Identify like terms.}\\
&2x^3-5x^2+11x-1\ & \ \ &\text{2) Combine like terms by adding/subtracting coefficients and write.}\\
&\ & \ \ &\text{in standard form.}\end{align}\)
&2x^3-5x^2+11x-1\ & \ \ &\text{2) Combine like terms by adding/subtracting coefficients and write.}\\
&\ & \ \ &\text{in standard form.}\end{align}\)
Example 2: Simplify \(\left(x^4+9x^3-7x-5\right)-\left(-2x^4+2x^2+5x-8\right)\).
\(\begin{align}&\left(x^4+9x^3-7x-5\right)+\left(2x^4-2x^2-5x+8\right)\ & \ \ &\text{1) Rewrite as addition by distributing a -1 into the second expression.}\\&\left(x^4+9x^3-7x-5\right)+\left(2x^4-2x^2-5x+8\right)\ & \ \ &\text{2) Identify like terms.}\\&3x^4+9x^3-2x^2-12x+3\ & \ \ &\text{3) Combine like terms by adding/subtracting coefficients and write}\\
&\ & \ \ &\text{in standard form.}\end{align}\)
&\ & \ \ &\text{in standard form.}\end{align}\)
Quick Check 1
Quick Check Solutions
- Simplify \(\left(-3x^5+2x^4+\frac{7}{2}x^2-1\right)+\left(-3x^4-x^3-5x^2+9\right)\)
- Simplify \(\left(7x-4x^2+3\right)-\left(x^3+7x^2-8x\right)\)
Quick Check Solutions
When multiplying polynomial expressions, each term of one polynomial expression must be distributed to each term in the other polynomial expression. You have done this in previous courses and in this course with quadratics, but it is always good to review it.
Example 3:
Example 3:
Example 4: Simplify \(\left(3x-1\right)\left(x+5\right)^2\).
Quick Check 2
Quick Check Solutions
- Simplify \(\left(2x^3-x^2+7\right)\left(x^2+5x+7\right)\)
- Simplify \(\left(2x-3\right)^3\)
Quick Check Solutions