Practice 2 Solutions

Determine if the given expressions will form an identity.

4 z^{2} + 4 z + 1 - 2 z + 2 4 z^{2} + 2 z + 3

4 z^{2} + 2 z - 1

No, this is not an identity.

x^{2} + 7 x + 10 - (x^{2} - 10 x + 21) x^{2} + 7 x + 10 - x^{2} + 10 x - 21 17 x - 11

(x - 1) (x + 7 - x + 10) + 6 (x - 1) (17) + 6 17 x - 17 + 6 17 x - 11

Yes, this is an identity.

$2 x^{2} y^{2} - 4 x^{2} - 9 y^{2}$ and $x^{2} (y + 2) (y - 2) + y^{2} (x + 3) (x - 3)$

2 x^{2} y^{2} - 4 x^{2} - 9 y^{2}

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

Yes, this is an identity.

a b + 3 a - b^{2} - 3 b

a b + 3 a - b^{2} - 3 b

Yes, this is an identity.

Note

The right side can be factored as $(a - b) (b + 3)$ . The left side can be factored as $(b + 3) (a - b)$ to show the identity.

(a + 2) (a + 2) (a + 2) (a^{2} + 4 a + 4) (a + 2) a^{3} + 2 a^{2} + 4 a^{2} + 8 a + 4 a + 8 a^{3} + 6 a^{2} + 12 a + 8

a^{3} - 2 a^{2} + 4 a + 2 a^{2} - 4 a + 8 + 6 a^{2} + 12 a a^{3} + 6 a^{2} + 12 a + 8

Yes, this is an identity.

x^{4} - 13 x^{2} + 36

(x^{2} - x - 6) (x^{2} - x - 6) x^{4} - x^{3} - 6 x^{2} - x^{3} + x^{2} + 6 x - 6 x^{2} + 6 x + 36 x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 36

No, this is not an identity.

Note

You may have factored to disprove the identity.

Find the missing value(s) of the given equation.

\begin{array}{rcl} 2 B x^{2} - B C x - 10 x + 5 C & = & 6 x^{2} - 11 x - 35 \\ 2 B x^{2} & = & 6 x^{2} \\ 2 B & = & 6 \\ B & = & 3 \end{array}

\begin{array}{rcl} 5 C & = & - 35 \\ C & = & - 7 \end{array}

\begin{array}{rcl} A^{3} x^{3} + B^{3} & = & 125 x^{3} + 64 \\ A^{3} & = & 125 \\ A & = & 5 \end{array}

\begin{array}{rcl} B^{3} & = & 64 \\ B & = & 4 \end{array}

\begin{array}{rcl} Q x^{2} - 4 x^{2} & = & 9 x^{2} \\ Q - 4 & = & 9 \\ Q & = & 13 \end{array}

\begin{array}{rcl} Y + 1 & = & 7 \\ Y & = & 6 \end{array}

\begin{array}{rcl} G x^{2} + 5 G x - 4 x - 20 - B & = & 3 x^{2} + 11 x + 8 \\ G x^{2} & = & 3 x^{2} \\ G & = & 3 \end{array}

\begin{array}{rcl} - 20 - B & = & 8 \\ - B & = & 28 \\ B & = & - 28 \end{array}

\begin{array}{rcl} 2 W x^{2} - 5 W x + 2 x - 5 + 3 x^{2} + 3 x - R x - R & = & 7 x^{2} - 7 x - 7 \\ 2 W x^{2} + 3 x^{2} & = & 7 x^{2} \\ 2 W + 3 & = & 7 \\ 2 W & = & 4 \\ W & = & 2 \end{array}

\begin{array}{rcl} - 5 - R & = & - 7 \\ R & = & 2 \end{array}

\begin{array}{rcl} 5 x^{2} - V x^{2} & = & 0 x^{2} \\ 5 - V & = & 0 \\ V & = & 5 \end{array}

\begin{array}{rcl} H x + H x & = & 14 \\ 2 H & = & 14 \\ H & = & 7 \end{array}