Problem 72 - A Graphical Approach to Precalculus with Limits

Question

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=2 x^{3}-3 x^{2}-5 x+6 ; \quad k=1$$

Step-by-Step Solution

Answer

The factorization of \(P(x)\) is \((x-1)(2x+3)(x-2)\).

Step 1: Use Synthetic Division

First, we will use the given zero \( k = 1 \) to perform synthetic division on \( P(x) = 2x^3 - 3x^2 - 5x + 6 \). Write down the coefficients [2, -3, -5, 6] and place \( k = 1 \) to the left in synthetic division. Proceed to carry down the 2, multiply and add through the row:\[\begin{array}{r|rrrr}1 & 2 & -3 & -5 & 6 \ & & 2 & -1 & -6 \\hline & 2 & -1 & -6 & 0 \\end{array}\]Since the remainder is 0, \( x - 1 \) is a factor of \( P(x) \), and the quotient is \( 2x^2 - x - 6 \).

Step 2: Factor the Quotient Quadratic

Now, factor the quadratic quotient \( 2x^2 - x - 6 \). We need two numbers that multiply to \(-12\) (\(2 \times -6\)) and add to \(-1\). These numbers are \(-4\) and \(3\). Rewrite the middle term and factor by grouping:\[2x^2 - 4x + 3x - 6 = (2x^2 - 4x) + (3x - 6) = 2x(x - 2) + 3(x - 2) = (2x + 3)(x - 2)\]Thus, the quadratic \(2x^2 - x - 6\) factors into \((2x + 3)(x - 2)\).

Step 3: Write the Complete Factorization

Since we've found that \( x - 1 \) is a factor, and the quadratic \(2x^2 - x - 6\) factors into \( (2x + 3)(x - 2) \), the complete factorization of \( P(x) \) is:\[ P(x) = (x - 1)(2x + 3)(x - 2) \].

Key Concepts

Synthetic Division

Synthetic division is a simplified method to divide a polynomial by a linear factor of the form \(x - k\). This technique significantly reduces the complexity and amount of work involved in polynomial division. For our exercise, we needed to factor the cubic polynomial \(P(x) = 2x^3 - 3x^2 - 5x + 6\), knowing that \(k = 1\) is a zero of the polynomial.To use synthetic division, you follow these steps:

• Write down the coefficients of the polynomial, in this case, [2, -3, -5, 6].
• Place the zero \(k\) to the left, in this case, 1.
• Carry down the leading coefficient (2) to the bottom row.
• Multiply the current number in the bottom row by \(k\), add this to the next coefficient above, and write the result in the bottom row.
• Repeat this process for each coefficient.

At the end of this process, the numbers in the bottom row give you the coefficients of the quotient polynomial, and the final number is the remainder. If the remainder is 0, \(x - k\) is indeed a factor of the polynomial as it is in our case, resulting in \(2x^2 - x - 6\).

Factorization

Factorization involves breaking down a complex expression into simpler "factors" that can be multiplied to obtain the original expression. In polynomial terms, this means expressing the polynomial as a product of its polynomial factors. For the given polynomial \(P(x) = 2x^3 - 3x^2 - 5x + 6\), after using synthetic division, we ended up with the quotient polynomial \(2x^2 - x - 6\).To factor this quadratic polynomial, we need:

• Two numbers whose product equals the product of the leading coefficient (2) and the constant term (-6), which is -12.
• The same two numbers must add up to the coefficient of the \(x\) term, which is -1 in this case.

The numbers found are 3 and -4. Rewriting the quadratic polynomial to factor by grouping, we get:

• Split the middle term: \(2x^2 - 4x + 3x - 6\).
• Group and factor: \(2x(x - 2) + 3(x - 2)\).
• Factor out the common term \((x - 2)\): \((2x + 3)(x - 2)\).

Thus, the quadratic factors into \((2x + 3)(x - 2)\), completing the factorization.

Zero of a Polynomial

The zero of a polynomial is a value of \(x\) that makes the polynomial equal to zero. In mathematical terms, if \(P(x)\) is a polynomial, then a number \(k\) is a zero if \(P(k) = 0\). This is critical because the zeros of a polynomial give you information about its factors and roots.For our problem, we began with the knowledge that \(k = 1\) is a zero of \(P(x) = 2x^3 - 3x^2 - 5x + 6\). This implies that \(x - 1\) is a factor of \(P(x)\) because substituting \(x = 1\) in the polynomial results in zero.Discovering a zero is paramount because:

• It helps to reduce the polynomial's degree when factored out.
• It gives insight into graphing the polynomial since these zeros are \(x\)-intercepts.
• Using the zero in synthetic division efficiently reveals remaining unfactored polynomial parts, leading to full factorization.

Therefore, identifying zeros not only aids in solving polynomial equations but also simplifies the understanding and manipulation of polynomial expressions.