Problem 3 - Algebra for College Students

Question

Factor completely each of the polynomials and indicate any that are not factorable using integers. $$x^{2}-11 x+28$$

Step-by-Step Solution

Answer

The polynomial factors to \((x - 4)(x - 7)\).

Step 1: Identify a Factoring Strategy

To factor a quadratic polynomial of the form \(x^2 + bx + c\), find two numbers that multiply to \(c\) (constant term) and add to \(b\) (coefficient of \(x\)). Here, we need two numbers that multiply to 28 and add up to 11.

Step 2: Find Numbers that Match Criteria

Test factor pairs of 28: (1, 28), (2, 14), (4, 7). The pair (4, 7) works as \(4 + 7 = 11\). These numbers satisfy both the sum and product requirements.

Step 3: Write Factored Form

Using the numbers 4 and 7, write the quadratic in its factored form: \((x - 4)(x - 7)\).

Step 4: Verify the Solution

Expand \((x - 4)(x - 7)\) to ensure it matches the original polynomial: \((x - 4)(x - 7) = x^2 - 7x - 4x + 28 = x^2 - 11x + 28\). The solution is correct.

Key Concepts

Factoring Strategy

When you approach a quadratic polynomial with the goal of factoring it, employing a factoring strategy is essential. A commonly used method for quadratic expressions of the form \(x^2 + bx + c\) is to find two numbers that simultaneously fulfill two conditions: they must multiply to \(c\) (the constant term) and add up to \(b\) (the coefficient of \(x\)). This process can involve:

• Identifying the constant term \(c\) and coefficient \(b\).
• Listing factor pairs of \(c\) and checking which pair sums to \(b\).

In the current example, our polynomial is \(x^2 - 11x + 28\). To factor this, we seek two numbers whose product is 28 (the constant term) and sum is -11 (the coefficient of \(x\)). You will find that this strategy reliably simplifies the process of factoring quadratics.

Quadratic Polynomials

Quadratic polynomials are mathematical expressions containing an unknown, typically denoted as \(x\), raised to the second power. These take the general form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The presence of the \(x^2\) term classifies these expressions as quadratics.

In the example given, \(x^2 - 11x + 28\) is our quadratic. It's considered a monic quadratic polynomial because the leading coefficient \(a\) is 1. Quadratics are fundamental due to their parabolic graphs and frequent occurrence in various fields like physics, engineering, and finance. Identifying the characteristics of a quadratic polynomial helps streamline the factoring process and aids in finding solutions efficiently.

Verification of Factored Form

After factoring a quadratic polynomial, it's crucial to check if the factored form is correct by verification. Verification involves expanding the factors back out to see if you recover the original polynomial. This ensures that the factorization is accurate and correct.

For our example, the polynomial \((x - 4)(x - 7)\) is proposed as the factored form of \(x^2 - 11x + 28\). To verify this:

• Begin by expanding the product: Use the distributive property to multiply \((x - 4)(x - 7)\).
• This results in \(x^2 - 7x - 4x + 28\).
• Simplify it to \(x^2 - 11x + 28\), which matches the original.

Conducting such a verification step is always advisable to ensure the accuracy of the factoring process.