Math / Algebra

QuestionFactoring and Applications Lesson \#4: Factori Factor. a) $4(3 x+1)^{2}-5(3 x+1)+1$

Studdy Solution

STEP 1

1. The expression is a quadratic in terms of $(3x + 1)$ .
2. We will use substitution to simplify factoring.
3. After substitution, we will factor the quadratic expression.

STEP 2

1. Substitute $u = 3x + 1$ to simplify the expression.
2. Rewrite the expression in terms of $u$ .
3. Factor the quadratic expression.
4. Substitute back to express the factorization in terms of $x$ .

STEP 3

Substitute $u = 3x + 1$ into the expression:
$4(3x + 1)^2 - 5(3x + 1) + 1$
becomes:
$4u^2 - 5u + 1$

STEP 4

Now, we have a quadratic expression in terms of $u$ :
$4u^2 - 5u + 1$

STEP 5

Factor the quadratic expression $4u^2 - 5u + 1$ .
To factor, look for two numbers that multiply to $4 \times 1 = 4$ and add to $-5$ . These numbers are $-4$ and $-1$ .
Rewrite the middle term using these numbers:
$4u^2 - 4u - u + 1$
Group terms:
$(4u^2 - 4u) - (u - 1)$
Factor each group:
$4u(u - 1) - 1(u - 1)$
Factor out the common factor $(u - 1)$ :
$(4u - 1)(u - 1)$

STEP 6

Substitute back $u = 3x + 1$ into the factored expression:
$(4(3x + 1) - 1)((3x + 1) - 1)$
Simplify each factor:
First factor:
$4(3x + 1) - 1 = 12x + 4 - 1 = 12x + 3$
Second factor:
$(3x + 1) - 1 = 3x$
Thus, the factored expression is:
$(12x + 3)(3x)$

Was this helpful?

QuestionFactoring and Applications Lesson \#4: Factori Factor. a) $4(3 x+1)^{2}-5(3 x+1)+1$

Studdy Solution

STEP 1

1. The expression is a quadratic in terms of $(3x + 1)$ .
2. We will use substitution to simplify factoring.
3. After substitution, we will factor the quadratic expression.

STEP 2

1. Substitute $u = 3x + 1$ to simplify the expression.
2. Rewrite the expression in terms of $u$ .
3. Factor the quadratic expression.
4. Substitute back to express the factorization in terms of $x$ .

STEP 3

Substitute $u = 3x + 1$ into the expression:
$4(3x + 1)^2 - 5(3x + 1) + 1$
becomes:
$4u^2 - 5u + 1$

STEP 4

Now, we have a quadratic expression in terms of $u$ :
$4u^2 - 5u + 1$

STEP 5

STEP 6

Substitute back $u = 3x + 1$ into the factored expression:
$(4(3x + 1) - 1)((3x + 1) - 1)$
Simplify each factor:
First factor:
$4(3x + 1) - 1 = 12x + 4 - 1 = 12x + 3$
Second factor:
$(3x + 1) - 1 = 3x$
Thus, the factored expression is:
$(12x + 3)(3x)$

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionFactoring and Applications Lesson \#4: Factori Factor. a) 4(3x+1)2−5(3x+1)+14(3 x+1)^{2}-5(3 x+1)+14(3x+1)2−5(3x+1)+1

Studdy Solution

STEP 1

1. The expression is a quadratic in terms of (3x+1) (3x + 1) (3x+1).2. We will use substitution to simplify factoring.3. After substitution, we will factor the quadratic expression.

STEP 2

1. Substitute u=3x+1 u = 3x + 1 u=3x+1 to simplify the expression.2. Rewrite the expression in terms of u u u.3. Factor the quadratic expression.4. Substitute back to express the factorization in terms of x x x.

STEP 3

Substitute u=3x+1 u = 3x + 1 u=3x+1 into the expression:4(3x+1)2−5(3x+1)+1 4(3x + 1)^2 - 5(3x + 1) + 1 4(3x+1)2−5(3x+1)+1becomes:4u2−5u+1 4u^2 - 5u + 1 4u2−5u+1

STEP 4

Now, we have a quadratic expression in terms of u u u:4u2−5u+1 4u^2 - 5u + 1 4u2−5u+1

STEP 5

STEP 6

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionFactoring and Applications Lesson \#4: Factori Factor. a) $4(3 x+1)^{2}-5(3 x+1)+1$

1. The expression is a quadratic in terms of $(3x + 1)$ .
2. We will use substitution to simplify factoring.
3. After substitution, we will factor the quadratic expression.

1. Substitute $u = 3x + 1$ to simplify the expression.
2. Rewrite the expression in terms of $u$ .
3. Factor the quadratic expression.
4. Substitute back to express the factorization in terms of $x$ .

Substitute $u = 3x + 1$ into the expression:
$4(3x + 1)^2 - 5(3x + 1) + 1$
becomes:
$4u^2 - 5u + 1$

Now, we have a quadratic expression in terms of $u$ :
$4u^2 - 5u + 1$