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Synthetic Division of Polynomials

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Synthetic Division of Polynomials

The volume of a rectangular prism is 2x^3 +  5x^2 - x - 6 . Determine if 2x + 3 is the length of one of the prism's sides.

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James Sousa: Polynomial Division: Synthetic Division

Guidance

Synthetic division is an alternative to long division from the previous concept. It can also be used to divide a polynomial by a possible factor, x - k . However, synthetic division cannot be used to divide larger polynomials, like quadratics, into another polynomial.

Example A

Divide 2x45x314x2+47x30 by x - 2 .

Solution: Using synthetic division, the setup is as follows:

To “read” the answer, use the numbers as follows:

Therefore, 2 is a solution, because the remainder is zero. The factored polynomial is 2x^3-x^2-16x+15 . Notice that when we synthetically divide by k , the “leftover” polynomial is one degree less than the original. We could also write (x-2)(2x^3-x^2-16x+15)=2x^4-5x^3-14x^2+47x-30 .

Example B

Determine if 4 is a solution to f(x)=5x^3+6x^2-24x-16 .

Using synthetic division, we have:

The remainder is 304, so 4 is not a solution. Notice if we substitute in x = 4 , also written f(4) , we would have f(4)=5(4)^3+6(4)^2-24(4)-16=304 . This leads us to the Remainder Theorem.

Remainder Theorem: If f(k) = r , then r is also the remainder when dividing by (x - k) .

This means that if you substitute in x = k or divide by k , what comes out of f(x) is the same. r is the remainder, but also is the corresponding y- value. Therefore, the point (k, r) would be on the graph of f(x) .

Example C

Determine if (2x - 5) is a factor of 4x^4-9x^2-100 .

Solution: If you use synthetic division, the factor is not in the form (x - k) . We need to solve the possible factor for zero to see what the possible solution would be. Therefore, we need to put \frac{5}{2} up in the left-hand corner box. Also, not every term is represented in this polynomial. When this happens, you must put in zero placeholders. In this example, we need zeros for the x^3- term and the x- term.

This means that \frac{5}{2} is a zero and its corresponding binomial, (2x - 5) , is a factor.

Intro Problem Revisit

If 2x + 3 divides evenly into 2x^3 +  5x^2 - x - 6 then it is the length of one of the prism's sides.

If we want to use synthetic division, notice that the factor is not in the form (x - k) . Therefore, we need to solve the possible factor for zero to see what the possible solution would be. If 2x+3=0 then x=-\frac{3}{2} . Therefore, we need to put \frac{-3}{2} up in the left-hand corner box.

When we perform the synthetic division, we get a remainder of 0. This means that (2x + 3) is a factor of the volume. Therefore, it is also the length of one of the sides of the rectangular prism.

Guided Practice

1. Divide x^3+9x^2+12x-27 by (x + 3) . Write the resulting polynomial with the remainder (if there is one).

2. Divide 2x^4-11x^3+12x^2+9x-2 by (2x + 1) . Write the resulting polynomial with the remainder (if there is one).

3. Is 6 a solution for f(x)=x^3-8x^2+72 ? If so, find the real-number zeros (solutions) of the resulting polynomial.

Answers

1. Using synthetic division, divide by -3.

The answer is x^2+6x-6-\frac{9}{x+3} .

2. Using synthetic division, divide by -\frac{1}{2} .

The answer is 2x^3-12x^2+18x-\frac{2}{2x+1} .

3. Put a zero placeholder for the x- term. Divide by 6.

The resulting polynomial is x^2-2x-12 . While this quadratic does not factor, we can use the Quadratic Formula to find the other roots.

x=\frac{2 \pm \sqrt{2^2-4(1)(-12)}}{2}=\frac{2 \pm \sqrt{4+48}}{2}=\frac{2 \pm 2 \sqrt{13}}{2}=1 \pm \sqrt{13}

The solutions to this polynomial are 6, 1+\sqrt{13} \approx 4.61 and 1-\sqrt{13} \approx -2.61 .

Vocabulary

Synthetic Division
An alternative to long division for dividing f(x) by k where only the coefficients of f(x) are used.
Remainder Theorem
If f(k) = r , then r is also the remainder when dividing by (x - k) .

Explore More

Use synthetic division to divide the following polynomials. Write out the remaining polynomial.

  1. (x^3+6x^2+7x+10) \div (x+2)
  2. (4x^3-15x^2-120x-128) \div (x-8)
  3. (4x^2-5) \div (2x+1)
  4. (2x^4-15x^3-30x^2-20x+42) \div (x+9)
  5. (x^3-3x^2-11x+5) \div (x-5)
  6. (3x^5+4x^3-x-2) \div (x-1)
  7. Which of the division problems above generate no remainder? What does that mean?
  8. What is the difference between a zero and a factor?
  9. Find f(-2) if f(x)=2x^4-5x^3-10x^2+21x-4 .
  10. Now, divide 2x^4-5x^3-10x^2+21x-4 by (x + 2) synthetically. What do you notice?

Find all real zeros of the following polynomials, given one zero.

  1. 12x^3+76x^2+107x-20; -4
  2. x^3-5x^2-2x+10; -2
  3. 6x^3-17x^2+11x-2; 2

Find all real zeros of the following polynomials, given two zeros.

  1. x^4+7x^3+6x^2-32x-32; -4, -1
  2. 6x^4+19x^3+11x^2-6x; 0, -2

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