Problem 49 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$9 y^{2}+13 y+4$$

Step-by-Step Solution

Answer

The factors of the polynomial \(9y^{2}+13y+4\) are \((9y+4)\) and \((y+1)\).

Step 1: Identify Polynomial Type

The given polynomial \(9y^{2}+13y+4\) is a quadratic polynomial because of its highest exponent, which is 2.

Step 2: Factor the polynomial

To factor the polynomial, we need to find two numbers that multiply to \(4*9 = 36\) and add up to \(13\). The numbers \(4\) and \(9\) do just that. However, we need to rearrange our polynomial first: rewrite \(13y\) as \(4y+9y\). Now, let's factorize step by step:\(9y^{2}+13y+4 = 9y^{2}+4y+9y+4 = y(9y+4)+1(9y+4) = (9y+4)(y+1)\)

Step 3: Check the factorization using multiplication

To ensure the solution is correct, we multiply our factors to see if it results into the original polynomial: \((9y+4)(y+1) = 9y^{2} +9y + 4y + 4 = 9y^{2}+ 13y+ 4\), yes, this results into our original polynomial, so the factorization is correct.

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials. This is similar to factoring numbers into their prime factors, but with algebraic expressions. By doing this, you can solve polynomial equations more easily. Factoring polynomials involves expressing a given polynomial as a product of two or more polynomials with smaller degrees. In our example, we started with a quadratic polynomial and eventually turned it into a product of two linear polynomials. It's important to check your factorization because only correct factors will satisfy the original polynomial equation when multiplied out as seen in the example:

• Identifying the polynomial type helps determine suitable methods for factorization.
• Always verify your end result by multiplying the factors back.

Quadratic Equation

A quadratic equation is any equation that can be represented in the standard form: \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). When graphed, such equations produce a curve known as a parabola. In our exercise, the polynomial \( 9y^2 + 13y + 4 \) is a quadratic equation because of the presence of the \(y^2\) term. Recognizing this equation's structure helps identify the approach for solving it, either by factoring, using quadratic formula, or completing the square. Quadratic equations are valuable not only in algebra but also in various real-world scenarios like physics, engineering, and economics.

Factoring Techniques

Factoring techniques are methods used to express polynomials as a product of simpler factors. Different techniques may be used based on the structure of the polynomial. For quadratic polynomials, common strategies include:

• Looking for patterns such as \(a^2 + 2ab + b^2 = (a+b)^2\).
• Finding pairs of numbers that multiply to the product of the leading coefficient and the constant term, then add up to the middle coefficient.
• Decomposing the middle term to arrange for easier grouping and factoring.

In the example of \(9y^2 + 13y + 4\), the middle term was rewritten as \(4y + 9y\), allowing factorization by grouping.

Graphing Utility

A graphing utility is a tool, often in the form of a calculator or software, used to visually represent equations. It helps confirm factors by showing whether the factorized form matches the original polynomial when graphed. For example, if you factor the quadratic \(9y^2 + 13y + 4\) into \((9y+4)(y+1)\) correctly, both forms should overlay each other perfectly on the graph. Graphing utilities enhance understanding by:

• Providing a visual check for polynomial factorization and solutions to polynomial equations.
• Allowing exploration of properties like roots, intercepts, and symmetry.

While powerful, these tools should complement, not replace, analytical methods.