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Use the image to answer the question.

x cubed plus 5 x squared plus 4 x plus 18 is divided by x plus 5 using the long division method.

Consider the polynomial division (x3+5x2+4x+18)÷(x+5)
. Give the quotient with the remainder expressed as a fraction.

9 answers

To perform polynomial long division for \( (x^3 + 5x^2 + 4x + 18) \div (x + 5) \), we follow these steps:

  1. Setup the long division:

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

  2. Multiply and subtract:

    • Multiply \( x^2 \) by \( x + 5 \):
      \( x^2 \cdot (x + 5) = x^3 + 5x^2 \).
    • Subtract from the original polynomial:
      \( (x^3 + 5x^2 + 4x + 18) - (x^3 + 5x^2) = 4x + 18 \).

  3. Divide the next term:

    • Divide \( 4x \) by \( x \) to get \( 4 \).

  4. Multiply and subtract again:

    • Multiply \( 4 \) by \( x + 5 \):
      \( 4 \cdot (x + 5) = 4x + 20 \).
    • Subtract from the current polynomial:
      \( (4x + 18) - (4x + 20) = -2 \).

At this point, we have:

  • Quotient: \( x^2 + 4 \)
  • Remainder: \( -2 \)

Thus, the final answer is:

\[ \text{Quotient} = x^2 + 4 \quad \text{with Remainder} = -2 \]

Expressing the remainder as a fraction gives:

\[ \text{Final Answer: } x^2 + 4 - \frac{2}{x + 5} \]

So the complete expression for the division is:

\[ x^2 + 4 - \frac{2}{x + 5} \]

se the image to answer the question.

x squared minus 4 is divided by x minus 1 using the long division method. The first expression of divisibility, left parenthesis x squared minus x right parenthesis, is subtracted from the dividend. The remainder is x minus 4. The quotient is x.

The image shows the first few steps of the polynomial division (x2−4)÷(x−1)
. Complete the polynomial division process, then give the quotient and the remainder expressed as a fraction.

(1 point)

To complete the polynomial long division for \( (x^2 - 4) \div (x - 1) \), we can follow these steps based on the provided information.

  1. Start the division:

    • The first term is \( x^2 \), and when divided by \( x \), the result is \( x \).

  2. Multiply and subtract:

    • Multiply \( x \) by \( (x - 1) \):
      \( x \cdot (x - 1) = x^2 - x \).
    • Subtract from the original polynomial:
      \( (x^2 - 4) - (x^2 - x) = -4 + x = x - 4 \).

  3. Now proceed with the new polynomial \( x - 4 \):

    • Divide \( x \) by \( x \) to get \( 1 \).

  4. Multiply and subtract again:

    • Multiply \( 1 \) by \( (x - 1) \):
      \( 1 \cdot (x - 1) = x - 1 \).
    • Subtract:
      \( (x - 4) - (x - 1) = -4 + 1 = -3 \).

Now we have our quotient and remainder:

  • Quotient: \( x + 1 \)
  • Remainder: \( -3 \)

Expressing the result, we write the full division as:

\[ x + 1 - \frac{3}{x - 1} \]

So the final answer is:

\[ \text{Quotient} = x + 1 \quad \text{with Remainder} = -3 \]

Complete expression for the division is:

\[ x + 1 - \frac{3}{x - 1} \]

When x3+2x+1
is divided by x+1
, the remainder is −2
. Apply the Remainder Theorem to find f(−1)
, where f(x)=x3+2x+1
.(1 point)
f(−1)=

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - c \), then the remainder of that division is equal to \( f(c) \).

In your case, you want to find \( f(-1) \) for the polynomial:

\[ f(x) = x^3 + 2x + 1 \]

Since you are dividing by \( x + 1 \), which can be rewritten as \( x - (-1) \), you are effectively evaluating \( f(-1) \).

According to the information provided, the remainder when dividing \( f(x) \) by \( x + 1 \) is \(-2\).

Thus:

\[ f(-1) = -2 \]

Therefore, the value is:

\[ f(-1) = -2 \]

Divide g(x)=3x2+4x−5
by x−7
to find g(7)
.(1 point)
g(7)=

To find \( g(7) \) using polynomial division of \( g(x) = 3x^2 + 4x - 5 \) by \( x - 7 \), we can use the Remainder Theorem, which states that \( g(c) \) is equal to the remainder when \( g(x) \) is divided by \( x - c \).

Let's divide \( g(x) \) by \( x - 7 \):

  1. Setting up the long division:

    • Divide the leading term \( 3x^2 \) by \( x \) to get \( 3x \).

  2. Multiply and subtract:

    • Multiply \( 3x \) by \( (x - 7) \):
      \[ 3x(x - 7) = 3x^2 - 21x. \]
    • Subtract from \( 3x^2 + 4x - 5 \): \[ (3x^2 + 4x - 5) - (3x^2 - 21x) = 4x + 21x - 5 = 25x - 5. \]

  3. Divide the next term:

    • Divide \( 25x \) by \( x \) to get \( 25 \).

  4. Multiply and subtract again:

    • Multiply \( 25 \) by \( (x - 7) \): \[ 25(x - 7) = 25x - 175. \]
    • Subtract this from the current polynomial: \[ (25x - 5) - (25x - 175) = -5 + 175 = 170. \]

Now we have completed the division:

  • Quotient: \( 3x + 25 \)
  • Remainder: \( 170 \)

Thus, \( g(7) \) is equal to the remainder when \( g(x) \) is divided by \( x - 7 \):

\[ g(7) = 170. \]

So, the final answer is:

\[ g(7) = 170. \]

Use the Factor Theorem to factor g(x)=x3−3x2−10x+24
into a product of linear factors, given that −3
is a zero of ​g(x)
.(1 point)
​g(x)=

To factor the polynomial \( g(x) = x^3 - 3x^2 - 10x + 24 \) using the Factor Theorem, given that \(-3\) is a zero of \(g(x)\), we know that \( (x + 3) \) is a factor of \(g(x)\).

Step 1: Perform Polynomial Division

We will divide \( g(x) \) by \( x + 3 \) using synthetic division:

  1. Write down the coefficients of \( g(x) \):

    • Coefficients: \( 1, -3, -10, 24 \)

  2. Set up synthetic division with \(-3\):

\[ \begin{array}{r|rrrr} -3 & 1 & -3 & -10 & 24 \ & & -3 & 18 & -6 \ \hline & 1 & -6 & 8 & 18 \ \end{array} \]

Step 2: Write Down the Result

After performing synthetic division, we obtain:

\[ g(x) = (x + 3)(x^2 - 6x + 8) \]

Step 3: Factor the Quadratic

Now, we need to factor the quadratic \( x^2 - 6x + 8 \):

  1. Find two numbers that multiply to \(8\) and add to \(-6\). These numbers are \(-2\) and \(-4\).

Thus, we can factor \( x^2 - 6x + 8 \) as:

\[ (x - 2)(x - 4) \]

Final Factorization

Combining everything together, we get:

\[ g(x) = (x + 3)(x - 2)(x - 4) \]

Conclusion

So the final factored form of \( g(x) \) is:

\[ \boxed{g(x) = (x + 3)(x - 2)(x - 4)} \]