Practice 2 Solutions

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

$|6 x| - 7 x^{3} + 10$

No, this is NOT a polynomial because the first term is an absolute value.

$5 x y + \sqrt{2} x^{2} + x^{2} y^{2}$

Yes, this is a 4th degree polynomial.
Third term: 2 + 2 = 4

Quartic trinomial

$\frac{1}{3} x^{5} - 4 x^{2}$

Yes, this is a 5th degree polynomial.
First term: 5

Quintic binomial

$\frac{3}{4 x} - \frac{1}{2} y$

No, this is NOT a polynomial. The first term has a variable in the denominator.

Simplify. Write answers in standard form.

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

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

$5 x^{2} + 35 x - 3 x^{3} + 6 x^{2}$

$- 3 x^{3} + 11 x^{2} + 35 x$

$(a + 2 b) (a^{2} - 2 a b + 4 b^{2})$

$a^{3} - 2 a^{2} b + 4 a b^{2} + 2 a^{2} b - 4 a b^{2} + 8 b^{3}$

$a^{3} + 8 b^{3}$

Note

You may notice this as a pattern for the sum of cubes.

$3 (5 r^{2} + 7 r v - 2 v^{2}) - 2 (2 r^{2} - r v)$

$15 r^{2} + 21 r v - 6 v^{2} - 4 r^{2} + 2 r v$

$11 r^{2} + 23 r v - 6 v^{2}$

$(n - 4) (n - 3) - (2 n + 1)$

$n^{2} - 7 n + 12 - 2 n - 1$

$n^{2} - 9 n + 11$

${(w + 1)}^{3}$

$(w + 1) (w + 1) (w + 1) (w^{2} + 2 w + 1) (w + 1) w^{3} + w^{2} + 2 w^{2} + 2 w + w + 1$

$w^{3} + 3 w^{2} + 3 w + 1$

Factor completely.

$9 n v + 24 n + 12 v + 32$

$3 n (v + 8) + 4 (v + 8)$

$(3 n + 4) (v + 8)$

$x^{4} - 16 y^{4}$

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

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

$8 x^{2} - 22 x - 21$

$(4 x + 3) (2 x - 7)$

$12 x^{3} y^{2} - 48 x^{2} y^{2} - 3 x - 12$

$12 x^{2} y^{2} (x - 4) - 3 (x - 4) (12 x^{2} y^{2} - 3) (x - 4) 3 (4 x^{2} y^{2} - 1) (x - 4)$

$3 (2 x y + 1) (2 x y - 1) (x - 4)$

$200 b^{2} + 120 b + 18$

$2 (100 b^{2} + 60 b + 9)$

$2 {(10 b + 3)}^{2}$

$36 x^{2} y^{2} - 15 x y - 6$

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

$3 (4 x y + 1) (3 x y - 2)$

Factor the sum or difference of cubes completely.

$27 x^{3} - 343$

$27 x^{3} = {(3 x)}^{3}, 343 = {(7)}^{3} (3 x - 7) ({(3 x)}^{2} + (3 x \cdot 7) + {(7)}^{2})$

$(3 x - 7) (9 x^{2} + 21 x + 49)$

$8 v^{3} - 125 m^{3}$

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

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

$2 b^{3} + 16 c^{3}$

$2 (b^{3} + 8 c^{3}); 8 c^{3} = {(2 c)}^{3} 2 (b + 2 c) (b^{2} - (b \cdot 2 c) + {(2 c)}^{2})$

$2 (b + 2 c) (b^{2} - 2 b c + 4 c^{2})$

$5 v^{3} + 625$

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

$5 (z + 5) (z^{2} - 5 z + 25)$