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Divide the polynomial 12x2+2x−9 by 4x−2 . Then express the polynomial as the product of the divisor and the quotient plus the remainder. (1 point)

Responses

12x2+2x−9=(4x−2)(3x+2)−13
12 x squared plus 2 x minus 9 is equal to open paren 4 x minus 2 close paren times open paren 3 x plus 2 close paren minus 13

12x2+2x−9=(4x−2)(3x+2)+5
12 x squared plus 2 x minus 9 is equal to open paren 4 x minus 2 close paren times open paren 3 x plus 2 close paren plus 5

12x2+2x−9=(4x−2)(3x+2)−5
12 x squared plus 2 x minus 9 is equal to open paren 4 x minus 2 close paren times open paren 3 x plus 2 close paren minus 5

12x2+2x−9=(4x−2)(3x−1)−11
12 x squared plus 2 x minus 9 is equal to open paren 4 x minus 2 close paren times open paren 3 x minus 1 close paren minus 11

9 answers

To divide the polynomial \( 12x^2 + 2x - 9 \) by \( 4x - 2 \), let's perform polynomial long division.

Step 1: Set up the division.

Dividing \( 12x^2 + 2x - 9 \) by \( 4x - 2 \).

Step 2: Divide the leading term.

  • Divide \( 12x^2 \) by \( 4x \) to get \( 3x \).

Step 3: Multiply and subtract.

  • Multiply \( 3x \) by \( 4x - 2 \): \[ 3x(4x - 2) = 12x^2 - 6x \]
  • Now subtract this from \( 12x^2 + 2x - 9 \): \[ (12x^2 + 2x - 9) - (12x^2 - 6x) = (2x + 6x) - 9 = 8x - 9 \]

Step 4: Divide the leading term again.

  • Now divide \( 8x \) by \( 4x \) to get \( 2 \).

Step 5: Multiply and subtract again.

  • Multiply \( 2 \) by \( 4x - 2 \): \[ 2(4x - 2) = 8x - 4 \]
  • Now subtract this from \( 8x - 9 \): \[ (8x - 9) - (8x - 4) = -9 + 4 = -5 \]

After performing the long division, we find:

  • Quotient: \( 3x + 2 \)
  • Remainder: \( -5 \)

Step 6: Express the result. Thus, we can express the original polynomial as: \[ 12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5 \]

The correct answer is: \[ 12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5 \]

So, the final response is: 12x² + 2x - 9 = (4x - 2)(3x + 2) - 5.

Use the image to answer the question.

A long division has for its divisor 3 x superscript 4 baseline plus 9 x squared minus 13. The dividend is x squared minus 5 x.

The image shows the first few steps of the polynomial division (3x4+9x2−13)÷(x2−5x) . Complete the division process and find the remainder of this polynomial division.

(1 point)
Responses

The remainder is 420x−13.
The remainder is 420 x minus 13 .

The remainder is 2,087.
The remainder is 2,087.

The remainder is 407x.
The remainder is 407 x .

The remainder is 407.

To find the remainder of the polynomial division \((3x^4 + 9x^2 - 13) \div (x^2 - 5x)\), we will carry out the long division process step by step.

Step 1: Setup the Division

  • Dividend: \(3x^4 + 0x^3 + 9x^2 - 13\)
  • Divisor: \(x^2 - 5x\)

Step 2: Divide the Leading Terms

Divide the leading term of the dividend by the leading term of the divisor: \[ \frac{3x^4}{x^2} = 3x^2 \] Write \(3x^2\) above the division line.

Step 3: Multiply and Subtract

Multiply \(3x^2\) by the entire divisor: \[ 3x^2 \cdot (x^2 - 5x) = 3x^4 - 15x^3 \]

Now, subtract this result from the original polynomial: \[ (3x^4 + 0x^3 + 9x^2 - 13) - (3x^4 - 15x^3) = 0x^4 + 15x^3 + 9x^2 - 13 \] This simplifies to: \[ 15x^3 + 9x^2 - 13 \]

Step 4: Repeat the Process

Now, divide the leading term of the new polynomial by the leading term of the divisor: \[ \frac{15x^3}{x^2} = 15x \] Write \(15x\) above the division line next to \(3x^2\).

Multiply \(15x\) by the divisor: \[ 15x \cdot (x^2 - 5x) = 15x^3 - 75x^2 \]

Subtract this from \(15x^3 + 9x^2 - 13\): \[ (15x^3 + 9x^2 - 13) - (15x^3 - 75x^2) = 0x^3 + 84x^2 - 13 \] This simplifies to: \[ 84x^2 - 13 \]

Step 5: Repeat Again

Divide the leading term of the new polynomial by the leading term of the divisor: \[ \frac{84x^2}{x^2} = 84 \] Write \(84\) above the line.

Multiply \(84\) by the divisor: \[ 84 \cdot (x^2 - 5x) = 84x^2 - 420x \]

Subtract this from \(84x^2 - 13\): \[ (84x^2 - 13) - (84x^2 - 420x) = 0x^2 + 420x - 13 \] This simplifies to: \[ 420x - 13 \]

Final Step: Determine Remainder

At this point, the degree of the remainder \(420x - 13\) is less than the degree of the divisor \(x^2 - 5x\). Thus, we can conclude that the remainder of the polynomial division is: \[ \text{Remainder} = 420x - 13 \]

Therefore, the correct response is: The remainder is \(420x - 13\).

Given f(x)=4x2−10x−3 , according to the Remainder Theorem, f(2) can be found by finding the remainder of which of the following division quotients?(1 point)
Responses

(4x2−10x−3)÷(x+2)
left parenthesis 4 x squared minus 10 x minus 3 right parenthesis divided by left parenthesis x plus 2 right parenthesis

(4x2−10x−3)÷x
left parenthesis 4 x squared minus 10 x minus 3 right parenthesis divided by x

(4x2−10x−3)÷2
left parenthesis 4 x squared minus 10 x minus 3 right parenthesis divided by 2

(4x2−10x−3)÷(x−2)

According to the Remainder Theorem, the value of a polynomial \( f(x) \) at a specific point \( r \), denoted \( f(r) \), can be found by dividing the polynomial by \( (x - r) \) and finding the remainder of that division.

In this case, we want to find \( f(2) \) for the polynomial \( f(x) = 4x^2 - 10x - 3 \). To apply the Remainder Theorem, we need to divide \( f(x) \) by \( (x - 2) \).

Therefore, the correct division quotient to find \( f(2) \) is:
\( (4x^2 - 10x - 3) \div (x - 2) \).

So the correct response is: (4x²−10x−3)÷(x−2).

Given the polynomial g(x) in its factored form as shown here: g(x)=(x−7)(3x−5) , apply the Factor Theorem. Which of the following is a zero of g(x) ?(1 point)
Responses

x=−5
x equals negative 5

x=−7
x equals negative 7

x=−53
x equals negative Start Fraction 5 over 3 End Fraction

x=7

The Factor Theorem states that if \( (x - r) \) is a factor of a polynomial \( g(x) \), then \( r \) is a zero of the polynomial \( g(x) \).

Given the factored form of the polynomial \( g(x) = (x - 7)(3x - 5) \):

  1. From the factor \( (x - 7) \), we see that \( 7 \) is a root because setting the factor equal to zero gives us: \[ x - 7 = 0 \implies x = 7 \]

  2. From the factor \( (3x - 5) \), we can find the other root by solving: \[ 3x - 5 = 0 \implies 3x = 5 \implies x = \frac{5}{3} \]

Now let's look at the response options provided:

  • \( x = -5 \)
  • \( x = -7 \)
  • \( x = -\frac{5}{3} \)
  • \( x = 7 \)

From our analysis, \( x = 7 \) is a zero of \( g(x) \).

Thus, the correct response is: \( x = 7 \).

Given a polynomial f(x) and that f(12)=0 , apply the Remainder Theorem and then determine which of the following is a factor of f(x) .(1 point)
Responses

(x−0)
left parenthesis x minus 0 right parenthesis

(x+12)
left parenthesis x plus 12 right parenthesis

(x−32)
left parenthesis x minus Start Fraction 3 over 2 End Fraction right parenthesis

(x−12)

According to the Remainder Theorem, if \( f(c) = 0 \) for a polynomial \( f(x) \), then \( (x - c) \) is a factor of the polynomial \( f(x) \).

Given that \( f(12) = 0 \), this means that \( 12 \) is a root of the polynomial. Therefore, we can conclude that:

\[ f(x) = (x - 12)g(x) \]

for some polynomial \( g(x) \), which indicates that \( (x - 12) \) is a factor of \( f(x) \).

Now, let's see which of the following options is a factor of \( f(x) \):

  • \( (x - 0) \)
  • \( (x + 12) \)
  • \( (x - \frac{3}{2}) \)
  • \( (x - 12) \)

The correct factor based on our analysis is: \( (x - 12) \).