Test 2 (Lessons 3–4): Working with Polynomials Solutions

1. Compare expressions A and B. Explain if each expression is or is not a polynomial.

1. $\left(–7x^2–8x^2+6\right)–\left(16x^3–3x^2+3x–7\right)$

$–7x^3–8x^2+6–16x^3+3x^2+7$

$–23x^3–5x^2–3x+13$

3. $\left(3x–5\right)\left(4x^2+x+1\right)$

$12x^3+3x^2+3x–20x^2–5x–5$

$12x^3–17x^2–2x–5$

4. ${\left(2x+7y\right)}^2$

$4x^2+14xy+14xy+49y^2$

$4x^2+28xy+49y^2$

5. $\left(5x^3+9x^{2}y–7x{y}^2\right)+\left(2x{y}^2–3x^{2}y–xy\right)$

$5x^3+6x^{2}y–5x{y}^2–xy$

6. $3x^2+7xy–20y^2$

$\left(3x–5y\right)\left(x+4y\right)$

7. $x^3+125$

$\left(x+5\right)\left(x^2–5x+25\right)$

8. Find the value of the unknown coefficient, P.
   $\left(P{x}^3+7xy–5y^2\right)–\left(6x^3–Pxy+4y^2\right)=–5x^3+8xy–9y^2$

$\begin{array}{rcl}P{x}^3–6x^3& =& –5x^3 OR 7xy–\left(–Pxy\right)=8xy\\ P–6& =& –5 7+P=8\\ P& =& 1 P=1\end{array}$

9. $4x^3–32y^3$

$4\left(x^3–8y^3\right)$

$4\left(x–2y\right)\left(x^2+2xy+4y^2\right)$

10. $x^4–81$

$\left(x^2–9\right)\left(x^2+9\right)$

$\left(x–3\right)\left(x+3\right)\left(x^2+9\right)$

11. Find the value of the unknown coefficients, M and N.
    $\left(Mx–8\right)\left(3x+N\right)=15x^2+11x–56$

$3M{x}^2+MNx–24x–8N=15x^2+11x–56$

$\begin{array}{rcl}3M{x}^2& =& 15x^2 –8N=–56\\ 3M& =& 15 \end{array}$

$\begin{array}{rcl}M& =& 5 N=7\end{array}$

12. Determine if a polynomial identity exists. Show and explain your work.

$$\begin{gathered} \left(x–4\right)\left(4x^2+20x+25\right) \\ 4x^3+20x^2+25x \\ \underline{+ –16x^2–80x–100} \\ 4x^3+ 4x^2–55x–100 \end{gathered}$$

$4x^3+4x^2–55x–100$
This is an identity because both sides of the equation are equal to one another.