Problem 58 - Algebra 1: Concepts and Skills

Question

Factor \(9 x^{2}-6 x-35.\) A. \((9 x-5)(x+7)\) B. \((3 x+5)(3 x-7)\) C. \((9 x+5)(x-7)\) D. \((3 x-5)(3 x+7)\)

Step-by-Step Solution

Answer

The factored form of the polynomial \(9 x^{2}-6 x-35\) is \((3 x-5)(3 x+7)\).

Step 1: Identify a, b, and c in the polynomial

In the equation \(9 x^{2}-6 x-35\), a=9, b=-6, and c=-35.

Step 2: Find two numbers that multiply to ac and add to b

We need to find two numbers that multiply to give the product \(ac = -315\) and add up to give the sum \(b=-6\). When we examine the factors of 315, we find that -15 and 21 multiply to give -315 and also add up to -6.

Step 3: Rewrite the middle term of the polynomial

The middle term of the polynomial is rewritten as the sum of the terms -15x and 21x, so we get \(9 x^{2}-15x+21x-35\).

Step 4: Factor by grouping

Group the terms to make factoring easier. This gives \((9 x^{2} - 15x) + (21x - 35)\). Then factor out the greatest common factor (GCF) from each group to get \(3x(3x-5)+7(3x-5)\).

Step 5: Write the final factored form

Now we can factor out the common binomial term \((3x-5)\) from the two terms in our previous step. We get that the original polynomial factors into \((3 x-5)(3 x+7)\).

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The highest exponent of the variable \(x\) is 2, which gives the equation a "quadratic" nature. Quadratic equations are key to many areas of mathematics and science due to their ability to model various phenomena.
Understanding the standard form \(ax^2 + bx + c\) is crucial, as it reveals how each term affects the shape of the graph, known as a parabola. The leading coefficient \(a\) influences the parabola's opening direction and width, while \(b\) shifts it along the x-axis. The constant \(c\) raises or lowers the graph vertically.

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

Quadratic equations can be solved using different methods such as factoring, completing the square, or using the quadratic formula \((-b \pm \sqrt{b^2 - 4ac})/(2a)\). This variety of methods allows flexibility depending on the specific problem structure.

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials with four terms. This technique involves grouping terms with common factors and then factoring those groups individually. It simplifies complex polynomials by breaking them down into smaller, more manageable parts.
In the exercise \(9x^2 - 6x - 35\), after identifying that \(-15\) and \(21\) can replace the middle term, we rewrite the polynomial as \(9x^2 - 15x + 21x - 35\). By grouping, we have two pairs: \((9x^2 - 15x)\) and \((21x - 35)\).
Now, factor each group:

• The first group \(9x^2 - 15x\) factors to \(3x(3x-5)\).
• The second group \(21x - 35\) factors to \(7(3x-5)\).

Once factoring by grouping is applied, look for a common binomial. Here, \(3x-5\) is the common binomial, allowing us to factor it out, resulting in a product of two binomials: \((3x-5)(3x+7)\). Factoring by grouping is particularly useful when the polynomial doesn't easily factor into a set of simple binomials from the outset.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest expression that divides all terms of a polynomial without leaving a remainder. Finding the GCF is an essential step in simplifying polynomials and can greatly assist in the factoring process.
When working on a problem like \(9x^2 - 15x + 21x - 35\), determining the GCF within each group helps isolate factors that can be combined. For \(9x^2 - 15x\), the common factor is \(3x\), since both terms can be divided by \(3x\). Similarly, for \(21x - 35\), the GCF is \(7\).

• Factor \(3x\) from \(9x^2 - 15x\): \(3x(3x - 5)\).
• Factor \(7\) from \(21x - 35\): \(7(3x - 5)\).

Once the GCFs are factored out, common factors across groups can reveal binomial expressions that simplify the entire polynomial. Mastering the GCF concept is crucial as it lays the groundwork for both simple and complex polynomial factorization.

Binomial Factoring

Binomial factoring involves breaking down a polynomial into two binomial expressions whose product gives the original polynomial. This technique simplifies expressions, making them easier to handle.
In our example, once factored by grouping, the common binomial term \(3x-5\) appears in both groups \(3x(3x-5)\) and \(7(3x-5)\). Recognizing this allows us to factor out \(3x-5\), pairing it with the remaining terms to form the complete factored expression:

• The factored form is \((3x-5)(3x+7)\).

This simplification occurs because binomial expressions, like \((3x - 5)\), behave as a single entity, just like variables. Becoming skilled at identifying and factoring by binomials can make solving quadratic equations more intuitive and less time-consuming. This approach is broadly applicable, especially when tackling standardized tests or real-world problems involving quadratic equations.