Questions: Use the long division to divide the polynomials. (5x^2-13x+3) / (x-2) Indicate the quotient, Q(x), and the remainder r(x), in the following equation: P(x)/d(x)=Q(x)+r(x)/d(x) where P(x)=5x^2-13x+3 and d(x)=x-2. Q(x)= r(x)=

Use the long division to divide the polynomials. (5x^2-13x+3) / (x-2) Indicate the quotient, Q(x), and the remainder r(x), in the following equation: P(x)/d(x)=Q(x)+r(x)/d(x) where P(x)=5x^2-13x+3 and d(x)=x-2. Q(x)= r(x)=
Transcript text: Use the long division to divide the polynomials. \[ \left(5 x^{2}-13 x+3\right) \div(x-2) \] Indicate the quotient, $Q(x)$, and the remainder $r(x)$, in the following equation: \[ \frac{P(x)}{d(x)}=Q(x)+\frac{r(x)}{d(x)} \] where $P(x)=5 x^{2}-13 x+3$ and $d(x)=x-2$. \[ Q(x)= \] $\square$ \[ r(x)= \] $\square$
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Solution

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Solution Steps

To divide the polynomial \( P(x) = 5x^2 - 13x + 3 \) by \( d(x) = x - 2 \) using long division, we follow these steps:

  1. Divide the leading term of the dividend \( 5x^2 \) by the leading term of the divisor \( x \) to get the first term of the quotient.
  2. Multiply the entire divisor by this term and subtract the result from the original polynomial.
  3. Repeat the process with the new polynomial obtained after subtraction until the degree of the remainder is less than the degree of the divisor.
  4. The quotient will be the result of the division, and the remainder will be the leftover polynomial.
Step 1: Polynomial Division

We are dividing the polynomial \( P(x) = 5x^2 - 13x + 3 \) by \( d(x) = x - 2 \) using long division. The first term of the quotient \( Q(x) \) is obtained by dividing the leading term of \( P(x) \) by the leading term of \( d(x) \): \[ \frac{5x^2}{x} = 5x \]

Step 2: Subtracting the Product

Next, we multiply the entire divisor \( d(x) \) by the first term of the quotient \( 5x \): \[ 5x \cdot (x - 2) = 5x^2 - 10x \] We then subtract this from \( P(x) \): \[ (5x^2 - 13x + 3) - (5x^2 - 10x) = -3x + 3 \]

Step 3: Continuing the Division

Now, we repeat the process with the new polynomial \( -3x + 3 \). The next term of the quotient is: \[ \frac{-3x}{x} = -3 \] We multiply the divisor by this term: \[ -3 \cdot (x - 2) = -3x + 6 \] Subtracting this from \( -3x + 3 \) gives: \[ (-3x + 3) - (-3x + 6) = -3 \]

Final Answer

The quotient \( Q(x) \) and the remainder \( r(x) \) can now be expressed as: \[ Q(x) = 5x - 3 \] \[ r(x) = -3 \] Thus, we can write the result of the division as: \[ \frac{P(x)}{d(x)} = Q(x) + \frac{r(x)}{d(x)} = (5x - 3) + \frac{-3}{x - 2} \]

The final answers are: \[ Q(x) = 5x - 3 \] \[ r(x) = -3 \]

\[ \boxed{Q(x) = 5x - 3} \] \[ \boxed{r(x) = -3} \]

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