Polynomials

High School Math 1 · Unit 3 · polynomial-operations · Teacher edition

polynomial-operations

Name ______________________________Date ______________

1.
Classify $4x^{3} - 2x + 7$ by its number of terms and its degree.
1. (A)
  binomial, degree $3$
2. (B)
  trinomial, degree $3$
3. (C)
  trinomial, degree $2$
4. (D)
  trinomial, degree $7$
2.
Expand and simplify $(2x - 3)(x + 5)$ .
Answer: ______________
3.
A rectangle has length $x + 6$ and width $x + 2$ . Its area, expanded, is $x^{2} + bx + c$ . What is $b$ ?
Answer: ______________
4.
Which is the expansion of $(x + 5)^{2}$ ?
1. (A)
  $x^{2} + 25$
2. (B)
  $x^{2} + 5x + 25$
3. (C)
  $x^{2} + 10x + 25$
4. (D)
  $x^{2} + 10x + 10$
5.
Real numbers $x$ and $y$ satisfy $x + y = 7$ and $xy = 10$ . What is $x^{2} + y^{2}$ ?
1. (A)
  $19$
2. (B)
  $29$
3. (C)
  $39$
4. (D)
  $49$
5. (E)
  $69$

Answer key — Polynomials

1.
(B)
trinomial, degree $3$
polynomial-operations-01
Three terms make a trinomial; the highest power is $x^3$ , so the degree is $3$ . Answer: trinomial, degree 3. ( $7$ is the constant, not the degree; degree $2$ would need an $x^2$ term.)
2.
$2x^2+7x-15$ polynomial-operations-02
$(2x - 3)(x + 5) = 2x^2 + 10x - 3x - 15 = \mathbf{2x^{2} + 7x - 15}$ .
3.
8polynomial-operations-03
$(x + 6)(x + 2) = x^2 + 2x + 6x + 12 = x^2 + 8x + 12$ , so $b = \mathbf{8}$ (and $c = 12$ ).
4.
(C)
$x^{2} + 10x + 25$
polynomial-operations-04
$(x + 5)^2 = x^2 + 5x + 5x + 25 = \mathbf{x^{2} + 10x + 25}$ . ( $x^2 + 25$ squares termwise and drops the middle; $x^2 + 5x + 25$ counts the cross product once; $x^2 + 10x + 10$ forgets to square the $5$ .)
5.
(B)
$29$
polynomial-operations-05
$x^2 + y^2 = (x + y)^2 - 2xy = 49 - 20 = \mathbf{29}$ . Check: $x = 5, y = 2$ gives $25 + 4 = 29$ . ( $49$ forgets to subtract $2xy$ ; $39$ subtracts $xy$ only once; $69$ adds $2xy$ instead of subtracting.)