Practice 1 Solutions

Determine if the following are polynomials. If so, classify by degree and number of terms.

$3 x^{2} y + 2 x y - 4$

Yes, this is a third degree polynomial.
1st term exp: 2 + 1 = 3
2nd term: 1 + 1 = 2
3rd term: 0

Cubic trinomial

Note

Q: How do you find the degree when the expression contains more than one variable?
A: Add the degree (exponents) of all of the variables.

$x^{4} y^{- 2} + 5 x + 1$

No, this is not a polynomial because the first term has a negative exponent.

$2 x^{2} + 4 x - \sqrt{x y}$

No, this is not a polynomial because xy is under a radical.

$x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 5 x + 1$

Yes, this is a 5th degree polynomial.
1st term: largest exponent: 5

Quintic polynomial with 6 terms

Simplify. Write answers in standard form.

$x^{2} (x y + 7) - (x + 1) (x - 1)$

$x^{3} y + 7 x^{2} - (x^{2} - 1) x^{3} y + 7 x^{2} - x^{2} + 1$

$x^{3} y + 6 x^{2} + 1$

Note

Q: What property allows you to multiply binomials together?
A: The distributive property

$(a b + 2) (a b - 3) - a b + 4$

$a^{2} b^{2} - 3 a b + 2 a b - 6 - a b + 4$

$a^{2} b^{2} - 2 a b - 2$

$(g + 5) (g^{2} - 3 g + 2)$

$g^{3} - 3 g^{2} + 2 g + 5 g^{2} - 15 g + 10$

$g^{3} + 2 g^{2} - 13 g + 10$

$2 x y (7 x^{2} y - 2) - 3 x (x^{2} y^{2} + 5)$

$14 x^{3} y^{2} - 4 x y - 3 x^{3} y^{2} - 15 x$

$11 x^{3} y^{2} - 4 x y - 15 x$

Note

Q: What is the degree of the simplified polynomial? Explain.
A: Degree 5, because $3 + 2 = 5$ .

$(k + 2) (k - 2) + (k + 3) (k - 3)$

$k^{2} - 4 + k^{2} - 9$

$2 k^{2} - 13$

$(2 r + 1) (r^{2} + 4 r - 2)$

$2 r^{3} + 8 r^{2} - 4 r + r^{2} + 4 r - 2$

$2 r^{3} + 9 r^{2} - 2$

Note

Q: Why is it not possible to combine terms with different degrees?
A: Because they are not like terms and only like terms can be added.

Factor completely.

Note

Problems 11–16
Q: What is the first step when factoring?
A: Determine if there is a GCF.

$3 x^{2} y^{2} - 5 x y - 12$

$(3 x y + 4) (x y - 3)$

$4 a^{2} b^{2} - 4 a^{2} - 9 b^{2} + 9$

$4 a^{2} (b^{2} - 1) - 9 (b^{2} - 1) (4 a^{2} - 9) (b^{2} - 1)$

$(2 a + 3) (2 a - 3) (b + 1) (b - 1)$

Note

Q: What special product occurs in this polynomial?
A: A difference of two squares

$5 m^{2} + 70 m + 245$

$5 (m^{2} + 14 m + 49)$

$5 {(m + 7)}^{2}$

Note

Q: What special product occurs in this polynomial?
A: A perfect square trinomial

$4 x^{2} - 26 x + 30$

$2 (2 x^{2} - 13 x + 15)$

$2 (2 x - 3) (x - 5)$

$35 q^{4} - 8 q^{3} - 3 q^{2}$

$q^{2} (35 q^{2} - 8 q - 3)$

$q^{2} (5 q + 1) (7 q - 3)$

$2 x y^{2} + 8 x - 9 y^{2} - 36$

$2 x (y^{2} + 4) - 9 (y^{2} + 4)$

$(2 x - 9) (y^{2} + 4)$

Factor the sum or difference of cubes completely.

Note

Problems 17–20
It may be helpful to write each term as a base to the 3rd power before using the factoring patterns for the sum and difference of cubes.

$8 y^{3} - 1$

$8 y = {(2 y)}^{3}, 1 = {(1)}^{3} (2 y - 1) ({(2 y)}^{2} + (2 y \cdot 1) + {(1)}^{2})$

$(2 y - 1) (4 y^{2} + 2 y + 1)$

$27 c^{3} + 64$

$27 c^{3} = {(3 c)}^{3}, 64 = {(4)}^{3} (3 c + 4) ({(3 c)}^{2} - (3 c \cdot 4) + {(4)}^{2})$

$(3 c + 4) (9 c^{2} - 12 c + 16)$

$250 v^{3} + 16$

$2 (125 v^{2} + 8); 125 v^{3} = {(5 v)}^{3}, 8 = {(2)}^{3} 2 (5 v + 2) ({(5 v)}^{2} - (5 v \cdot 2) + {(2)}^{2})$

$2 (5 v + 2) (25 v^{2} - 10 v + 4)$

$x^{3} y^{3} - a^{3} b^{3}$

$x^{3} y^{3} = {(x y)}^{3}, a^{3} b^{3} = {(a b)}^{3} (x y - a b) ({(x y)}^{2} + (x y \cdot a b) + {(a b)}^{2})$

$(x y - a b) (x^{2} y^{2} + a b x y + a^{2} b^{2})$