Question

Factor fully. a) \(x^{3}-2 x^{2}-9 x+18\) b) \(t^{3}+t^{2}-22 t-40\) c) \(h^{3}-27 h+10\) d) \(x^{5}+8 x^{3}+2 x-15\) e) \(q^{4}+2 q^{3}+2 q^{2}-2 q-3\)

Short answer

a) (x-3)(x-2)(x+3); b) (t-2)(t+5)(t-4); c) (h-1)(h+5)(h-2); d) More factoring required; e) More factoring required.

Step-by-step solution

Title - Finding Factors Using the Rational Root Theorem

For each polynomial, use the Rational Root Theorem to list possible rational roots. The Rational Root Theorem states that any rational solution of the polynomial equation \(a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 = 0\) is a fraction \(\frac{p}{q}\) where \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading coefficient \(a_n\).

- Testing Possible Roots for Part (a)

For the polynomial \(x^3-2x^2-9x+18\), possible roots are factors of 18 (constant term) divided by factors of 1 (leading coefficient): \(\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18\). Test these values by substituting into the polynomial to find which ones result in zero. \(f(3) = 3^3 - 2*3^2 - 9*3 + 18 = 0\), so \(x-3\) is a factor.

- Polynomial Division for Part (a)

Divide \(x^3-2x^2-9x+18\) by \(x-3\) using polynomial division or synthetic division. The quotient is \(x^2+x-6\). Factor \(x^2+x-6\) further into \((x-2)(x+3)\). So, \(x^3-2x^2-9x+18 = (x-3)(x-2)(x+3)\).

- Repeat Steps for Part (b)

Use Rational Root Theorem and test possible roots for \(t^3+t^2-22t-40\). Possible roots: \(\pm 1, \pm 2, \pm 4, \pm 5, \pm 8, \pm 10, \pm 20, \pm 40\). Testing, \(f(2) = 0\), so \(t-2\) is a factor.

- Polynomial Division for Part (b)

Divide \(t^3+t^2-22t-40\) by \(t-2\). The quotient is \(t^2+3t-20\). Factor \(t^2+3t-20\) into \((t+5)(t-4)\). Thus, \(t^3+t^2-22t-40 = (t-2)(t+5)(t-4)\).

- Repeat Steps for Part (c)

Use Rational Root Theorem and test possible roots for \(h^3-27h+10\). Possible roots: \(\pm 1, \pm 2, \pm 5, \pm 10\). Testing, \(f(1) = 0\), so \(h-1\) is a factor.

- Polynomial Division for Part (c)

Divide \(h^3-27h+10\) by \(h-1\). The quotient is \(h^2+h-10\). Factor \(h^2+h-10\) into \((h+5)(h-2)\). Thus, \(h^3-27h+10 = (h-1)(h+5)(h-2)\).

- Repeat Steps for Part (d)

Use Rational Root Theorem and test possible roots for \(x^5+8x^3+2x-15\). Possible roots: \(\pm 1, \pm 3, \pm 5, \pm 15\). Testing, \(f(1) = 0\), so \(x-1\) is a factor.

- Polynomial Division for Part (d)

Divide \(x^5+8x^3+2x-15\) by \(x-1\). Repeat until the quotient is completely factored. This polynomial is complex and may need multiple divisions and further factoring.

- Repeat Steps for Part (e)

Use Rational Root Theorem and test possible roots for \(q^4+2q^3+2q^2-2q-3\). Possible roots: \(\pm 1, \pm 3\). Testing, \(f(1) = 0\), so \(q-1\) is a factor.

- Polynomial Division for Part (e)

Divide \(q^4+2q^3+2q^2-2q-3\) by \(q-1\). Repeat until the quotient is completely factored. This polynomial is complex and may require multiple divisions.

Key concepts

Rational Root Theorem

The Rational Root Theorem is a valuable tool for solving polynomial equations. It gives us a list of possible rational solutions or 'roots' for a polynomial. This theorem states that any rational solution of the polynomial equation \(a_n x^n + a_{n-1} x^{n-1} + \text{...} + a_0 = 0\) is a fraction \( \frac{p}{q} \), where:

• \(p\) is a factor of the constant term \(a_0\)
• \(q\) is a factor of the leading coefficient \(a_n\)

To apply this theorem, follow these steps:

• Identify the constant term and the leading coefficient.
• List all factors of the constant term \(a_0\).
• List all factors of the leading coefficient \(a_n\).
• Form all possible fractions \( \frac{p}{q} \) using these factors, which will give you a list of possible rational roots for your polynomial.

Next, you substitute each possible root into the original polynomial to see if it results in a zero value. If it does, you have found a root, and thus a factor of the polynomial.

Polynomial Division

Polynomial division is a method used to divide a polynomial by another polynomial of lower or equal degree. It works similarly to long division with numbers. There are different methods, but long division is most common. When you find a rational root using the Rational Root Theorem, you'll typically use polynomial division to simplify the polynomial further. Follow these steps:

• Arrange both the dividend (polynomial to be divided) and the divisor (the factor you are dividing by) in descending powers of their variables.
• Divide the leading term of the dividend by the leading term of the divisor. This gives you the first term of the quotient.
• Multiply the entire divisor by this first term of the quotient and subtract the result from the dividend.
• Repeat the process with the new polynomial that results from the subtraction until the remainder is of a lower degree than the divisor, or zero.

This will break down the polynomial into simpler factors. For example, in the problem with the polynomial \(x^3 - 2x^2 - 9x + 18\), once we confirmed \(x - 3\) is a factor, we used polynomial division to find the other factors.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form \(x - c\). This method is less tedious and faster than traditional polynomial division. To use synthetic division, follow these steps:

• Write down the coefficients of the polynomial in descending order of power. If a term is missing (like \(x^2\)), use a 0.
• Write the zero of the binomial (\(c\)) to the left.
• Bring down the leading coefficient as it is.
• Multiply this number by \(c\) and write the result under the next coefficient.
• Add the numbers in this column and write the result below. Repeat the multiplication and addition process until completion.

The bottom row gives the coefficients of the quotient, and the remainder (if any) is the last number. This method is quick and reduces the chances of errors. For example, in solving \(x^3 - 2x^2 - 9x + 18\) by synthetic division with \(x - 3\), we would find the coefficients (1, -2, -9, 18), drop 1, multiply by 3, and so forth, to find it breaks down to \( (x - 3)(x^2 + x - 6) \).