Factoring Polynomials How To Factor F(x) = 4x³ + 7x² - 62x + 15

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Hey everyone! Today, we're diving deep into the world of polynomial factorization, and we're going to tackle a pretty interesting problem. We've got the cubic polynomial f(x) = 4x³ + 7x² - 62x + 15, and we know that -5 is one of its zeros. Our mission, should we choose to accept it (and we do!), is to factor this polynomial into linear factors. This means we want to express f(x) as a product of terms like (x - a), where a is a root of the polynomial. Buckle up, guys, because we're about to embark on a mathematical adventure!

Understanding Polynomial Factorization

Before we jump straight into the solution, let's take a moment to understand polynomial factorization better. At its core, factoring a polynomial is like reverse multiplication. Think about it: when you multiply two or more polynomials together, you get another polynomial. Factoring is the process of taking that resulting polynomial and breaking it back down into its original factors. These factors are usually smaller polynomials, and in our case, we're aiming for linear factors, which are polynomials of degree one (like x - a).

Why is factoring so important, you ask? Well, it opens the door to solving polynomial equations. If we can factor a polynomial f(x) into linear factors, we can easily find its zeros, which are the values of x that make f(x) = 0. These zeros are also the x-intercepts of the polynomial's graph, so factoring gives us valuable information about the polynomial's behavior. In many science and engineering fields, equations are expressed in polynomial form so polynomial factorization is an essential tool to handle this.

The Factor Theorem: Our Guiding Light

The Factor Theorem is a crucial concept that guides our factoring journey. It states that if f(a) = 0 for some value a, then (x - a) is a factor of the polynomial f(x). Conversely, if (x - a) is a factor of f(x), then f(a) = 0. This theorem is our secret weapon because it connects zeros and factors, giving us a direct way to find factors if we know the zeros (or vice versa). Also, remember the Remainder Theorem is important, which can help us find the remainder when f(x) is divided by (x - a), giving us some hints for polynomial factorization.

In our problem, we're given that -5 is a zero of f(x) = 4x³ + 7x² - 62x + 15. This means that f(-5) = 0, and according to the Factor Theorem, (x - (-5)), which simplifies to (x + 5), must be a factor of f(x). This is our starting point, guys! We know one factor already, and that's a huge step forward.

Step-by-Step Solution: Factoring f(x) = 4x³ + 7x² - 62x + 15

Okay, let's get down to business and factor this polynomial. We know that (x + 5) is a factor, so we can use polynomial long division or synthetic division to divide f(x) by (x + 5). This will give us another factor, which will be a quadratic polynomial (degree 2). We can then factor the quadratic to get the remaining linear factors.

1. Polynomial Long Division or Synthetic Division

Let's use synthetic division, as it's a bit more streamlined for this process. Synthetic division is a shorthand method for dividing a polynomial by a linear factor of the form (x - a). Here's how it works:

  1. Write down the coefficients of the polynomial: 4, 7, -62, 15.
  2. Write down the zero we're dividing by (-5) to the left.
  3. Bring down the first coefficient (4) below the line.
  4. Multiply the number we just brought down (4) by the zero (-5), and write the result (-20) below the second coefficient (7).
  5. Add the second coefficient (7) and the result (-20) to get -13, and write it below the line.
  6. Multiply -13 by -5 to get 65, and write it below -62.
  7. Add -62 and 65 to get 3, and write it below the line.
  8. Multiply 3 by -5 to get -15, and write it below 15.
  9. Add 15 and -15 to get 0, which is the remainder. This confirms that (x + 5) is indeed a factor!

The numbers below the line (4, -13, 3) are the coefficients of the quotient polynomial. Since we divided a cubic polynomial by a linear factor, the quotient will be a quadratic. So, the quotient is 4x² - 13x + 3.

2. Factoring the Quadratic

Now we have f(x) = (x + 5)(4x² - 13x + 3). Our next step is to factor the quadratic 4x² - 13x + 3. There are several ways to do this, including:

  • Factoring by grouping: This involves finding two numbers that multiply to (4 * 3 = 12) and add up to -13. These numbers are -12 and -1. We can then rewrite the middle term and factor by grouping.
  • Using the quadratic formula: If the quadratic doesn't factor easily, we can use the quadratic formula to find its roots, and then write it in factored form.

Let's use factoring by grouping here. We rewrite the quadratic as 4x² - 12x - x + 3. Now we can factor by grouping:

  • 4x² - 12x - x + 3 = 4x(x - 3) - 1(x - 3)
  • =(4x - 1)(x - 3)

So, the quadratic 4x² - 13x + 3 factors into (4x - 1)(x - 3).

3. Putting It All Together

We've now factored both parts of our polynomial. We have:

  • f(x) = (x + 5)(4x² - 13x + 3)
  • f(x) = (x + 5)(4x - 1)(x - 3)

And there you have it, guys! We've successfully factored the cubic polynomial f(x) = 4x³ + 7x² - 62x + 15 into linear factors: (x + 5)(4x - 1)(x - 3).

Finding the Zeros

Now that we have the factored form, finding the zeros is a breeze. The zeros are the values of x that make f(x) = 0. Since f(x) is a product of linear factors, it will be zero if any of those factors are zero. So, we set each factor equal to zero and solve for x:

  • x + 5 = 0 => x = -5
  • 4x - 1 = 0 => x = 1/4
  • x - 3 = 0 => x = 3

Therefore, the zeros of f(x) are -5, 1/4, and 3. These are the points where the graph of f(x) crosses the x-axis.

Visualizing the Polynomial

It's always helpful to visualize what we've done. The graph of f(x) = 4x³ + 7x² - 62x + 15 is a cubic curve that crosses the x-axis at x = -5, x = 1/4, and x = 3. The factored form of the polynomial gives us this information directly, which is one of the many benefits of factoring.

Common Mistakes to Avoid

Factoring polynomials can be tricky, so let's talk about some common mistakes to watch out for:

  • Forgetting to factor completely: Make sure you factor the polynomial as much as possible. In our case, we needed to factor the quadratic after dividing by the linear factor.
  • Making sign errors: Pay close attention to the signs when using synthetic division or factoring by grouping. A small sign error can throw off the entire solution.
  • Incorrectly applying the quadratic formula: Double-check your values and calculations when using the quadratic formula.
  • Not verifying your answer: After factoring, you can always multiply the factors back together to make sure you get the original polynomial. This is a great way to catch any errors.

Conclusion: Mastering Polynomial Factorization

So, there you have it! We've successfully factored the cubic polynomial f(x) = 4x³ + 7x² - 62x + 15 into linear factors using the Factor Theorem, synthetic division, and factoring by grouping. We also found the zeros of the polynomial and discussed some common mistakes to avoid. Polynomial factorization is a fundamental skill in algebra, and mastering it will open doors to more advanced mathematical concepts.

Remember, the key to success in math is practice, practice, practice! The more you factor polynomials, the better you'll become at it. So, keep those pencils sharp and keep exploring the wonderful world of mathematics, guys! You've got this!