Chapter 6: Q 108. (page 589)

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error.
$- 81 a^{2} + 153 a + 18$

Short Answer

Expert verified

The solution is $\begin{array}{rcl} - 9 (a - 2) (9 a + 1) \end{array}$ .

Step by step solution

Step 1. Given information

Consider the trinomial.

$- 81 a^{2} + 153 a + 18$

Step 2. Write the trinomial in descending order and then find the greatest common factor.

The trinomial $- 81 a^{2} + 153 a + 18$ is given in descending order.

There is a greatest common factor in the given trinomial.

Factor the greatest common factor.

$- 81 a^{2} + 153 a + 18 = - 9 (9 a^{2} - 17 a - 2)$

Step 3. Find the factors of the first term and the last term of the trinomial 9a2-17a-2.

The first term of the trinomial $9 a^{2} - 17 a - 2$ is $9 a^{2}$ .

So, the factors of $9 a^{2}$ are as follows:

$\begin{array}{rcl} 9 a^{2} & = & a \cdot 9 a \\ 9 a^{2} & = & 3 a \cdot 3 a \end{array}$

The last term of the trinomial $9 a^{2} - 17 a - 2$ is $- 2$ .

Since the last term of the polynomial $9 a^{2} - 17 a - 2$ is negative, the factors of the last term will have opposite signs.

The factors of $- 2$ are as follows:

$\begin{array}{rcl} - 2 & = & 1 \cdot (- 2) \\ - 2 & = & (- 1) \cdot 2 \end{array}$

Step 4. Make a table for all the combination of factors of 9a2-17a-2.

Possible factors	Product
$(a + 1) (9 a - 2)$	$9 a^{2} + 7 a - 2$
$(a - 2) (9 a + 1)$	$9 a^{2} - 17 a - 2$
$(a - 1) (9 a + 2)$	$9 a^{2} - 7 a - 2$
$(a + 2) (9 a - 1)$	$9 a^{2} + 17 a - 2$
role="math" localid="1645098383241" $(3 a + 1) (3 a - 2)$	$9 a^{2} - 3 a - 2$
role="math" localid="1645098395490" $(3 a - 1) (3 a + 2)$	$9 a^{2} + 3 a - 2$

From the table, conclude that the combination $(a - 2) (9 a + 1)$ is correct.

Step 5. Check by multiplying all the factors -9(a-2)(9a+1).

$\begin{array}{rcl} - 9 (a - 2) (9 a + 1) & = & - 9 (9 a^{2} + a - 18 a - 2) \\ = & - 9 (9 a^{2} - 17 a - 2) \\ = & - 81 a^{2} + 153 a + 18 \end{array}$

Hence, the factor is $\begin{array}{rcl} - 9 (a - 2) (9 a + 1) \end{array}$ .

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Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error.
$- 81 a^{2} + 153 a + 18$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Write the trinomial in descending order and then find the greatest common factor.

Step 3. Find the factors of the first term and the last term of the trinomial 9a2-17a-2.

Step 4. Make a table for all the combination of factors of 9a2-17a-2.

Step 5. Check by multiplying all the factors -9(a-2)(9a+1).

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Factor Trinomials of the Form ax2+bx+cUsing Trial and Error.−81a2+153a+18

Short Answer

Step by step solution

Step 1. Given information

Step 2. Write the trinomial in descending order and then find the greatest common factor.

Step 3. Find the factors of the first term and the last term of the trinomial 9a2-17a-2.

Step 4. Make a table for all the combination of factors of 9a2-17a-2.

Step 5. Check by multiplying all the factors -9(a-2)(9a+1).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Discrete Mathematics

Mechanics Maths

Decision Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Factor Trinomials of the Form $a x^{2} + b x + c$ Using Trial and Error.
$- 81 a^{2} + 153 a + 18$