Problem 29 - College Algebra and Calculus: An Applied Approach

Question

Factor the trinomial.\(y^{2}+y-20\)

Step-by-Step Solution

Answer

The factored form of the trinomial is \(y+5\)\(y-4\).

Step 1: Identify A, B, and C values

In the trinomial expression \(y^{2}+y-20\), we identify the A, B and C values from the standard quadratic equation format \(Ax^{2}+Bx+C\). Here, A=1, B=1, and C=-20.

Step 2: Find factors of AC

Find two numbers that are factors of A*C (1*-20=-20) and add up to B=1. The numbers are +5 and -4.

Step 3: Write the factors

Write the trinomial in terms of two binomials. We use the numbers found in step 2 (5 and -4), in our binomials. Our final result is \(y+5\)\(y-4\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that takes the form of \(Ax^2 + Bx + C = 0\), where \(A\), \(B\), and \(C\) are constants, and \(x\) represents an unknown variable. The key feature of quadratic equations is that the highest exponent of the variable \(x\) is 2, which gives it the name 'quadratic', from the Latin word 'quadratus', meaning 'square'.

Quadratic equations can often have two solutions for the variable \(x\), which can be found by factoring, completing the square, or using the quadratic formula. Factoring is a method that involves breaking down the expression into simpler binomial expressions that are easier to solve. Once these expressions are set equal to zero, you can find the values of \(x\) that satisfy the equation. This is an important step in solving real-world problems where quadratic equations naturally arise, like calculating projectile trajectories or determining certain economic outcomes.

Trinomial Factoring

Trinomial factoring is a specific technique used to simplify polynomial expressions, particularly those that are quadratic in nature. In the expression \(y^2 + y - 20\), you are dealing with a trinomial that needs to be expressed as a product of two binomials.

The process involves:

• Identifying the coefficient values \(A\), \(B\), and \(C\) from the expression.
• Finding two numbers that multiply to \(A\times C\) and add up to \(B\).
• Rewriting the middle term using these two numbers, which helps in grouping and eventually factoring by grouping or substitution.

In the example \(y^2 + y - 20\):

• Your \(A\) value is 1, \(B\) is 1, and \(C\) is -20.
• You need two numbers whose product is \(-20\) and sum is \(1\); the correct pair is \(+5\) and \(-4\).

The trinomial then factors to \((y + 5)(y - 4)\), which can be verified by expanding the expression back to check for correctness.

Polynomial Expressions

Polynomial expressions are sums of various power terms of a variable, each multiplied by their individual coefficients. They are fundamental building blocks in algebra and can range from simple expressions like \(2x + 3\) to more complex ones such as \(4x^3 - x^2 + 6x - 8\).

A polynomial expression is said to be of degree \(n\) if the highest power of the variable \(x\) is \(n\). They can be of various types depending on their degree:

• Linear Polynomial: Degree 1, like \(3x + 1\).
• Quadratic Polynomial: Degree 2, like \(x^2 + 5x + 6\).
• Cubic Polynomial: Degree 3, like \(2x^3 - x^2 + 1\).

Factoring polynomial expressions is useful for simplifying them or solving polynomial equations. The process requires identifying parts of the expression that can be combined or restructured to reveal patterns or common factors, often turning polynomials into products of their simpler roots. This is particularly advantageous when dealing with quadratic or higher-degree polynomials to determine solutions or to simplify expressions for further mathematical operations.