Problem 56 - College Algebra and Calculus: An Applied Approach

Question

Completely factor the expression.\(3 x^{4}+x^{3}-10 x^{2}\)

Step-by-Step Solution

Answer

The completely factored form of the expression \(3x^{4} + x^{3} -10x^{2}\) is \(x^{2}(3x-5)(x+2)\).

Step 1: STEP 1: Identify the GCF

Observe all terms in the polynomial \(3x^{4} + x^{3} -10x^{2}\). The greatest common factor (GCF), which every term in the polynomial has, is \(x^{2}\).

Step 2: STEP 2: Factor out the GCF

The GCF is factored out from every term in the polynomial by dividing each term by the GCF \(x^{2}\). This results in \(x^{2}(3x^{2} + x - 10)\).

Step 3: STEP 3: Factor the quadratic polynomial

Now, factorize the quadratic expression \(3x^{2} + x - 10\) inside the parentheses. This can be factored as \((3x-5)(x+2)\).

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is a key concept in polynomial factorization. It refers to the largest factor shared by all terms in a polynomial. Identifying the GCF is the first and often the easiest step in simplifying a polynomial expression.
For the expression provided, \(3x^4 + x^3 - 10x^2\), we need to look at each term to find a common factor. Here, the greatest power of \(x\) that appears in every term is \(x^2\).

• First term: \(3x^4\) – contains \(x^2\)
• Second term: \(x^3\) – contains \(x^2\)
• Third term: \(-10x^2\) – contains \(x^2\)

Taking \(x^2\) as a common factor simplifies the polynomial by grouping the remaining expression inside the parentheses. This step not only reduces the expression size but also helps in simplifying further factorization steps.

Quadratic Polynomial

Understanding quadratic polynomials is crucial for effective factorization. A quadratic polynomial is an expression in the form of \(ax^2 + bx + c\). In the given problem, after factoring out the GCF \(x^2\), we are left with the quadratic polynomial \(3x^2 + x - 10\).
Quadratic polynomials often have two roots, and can be rewritten as a product of two binomial expressions: \((px + q)(rx + s)\). Finding these factors requires examining how the numbers multiply and add up.

• Coefficients multiplication: \(p \times r = a\)
• Cross-multiplication that sums up to the middle term coefficient: \(q \times s + p \times r = b\)

Identifying these factors allows us to express the quadratic equation fully factored, aiding in solving the polynomial and understanding how the expression behaves across its roots.

Factoring Techniques

Factoring techniques play a vital role in breaking down polynomials into their simpler parts. Various methods are used based on the degree and nature of the polynomial.
In the case of \(3x^2 + x - 10\), the technique involves splitting the middle term or trial and error to find the set of two numbers that, when multiplied, yield the product of the leading coefficient and the constant term, \(3 \times -10 = -30\). Simultaneously, these numbers must add to \(1\), the middle coefficient.

• Split: Look for two numbers that, when combined, give the middle term coefficient when appropriately distributed.
• Check: The multiplication results must align with the initial constant and leading term multiplication.

The correct factorization of the quadratic polynomial \(3x^2 + x - 10\) using these steps gives us \((3x - 5)(x + 2)\). An effectively broken down expression enables easier problem solving and deeper comprehension of polynomial behavior.