91Ó°ÊÓ

Understanding polynomials is a key part of algebra. A polynomial is an expression made up of variables, coefficients, and exponents. The general form is something like: \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\). Each part of a polynomial is called a term.
Polynomials can have constants (like 3 or -2), variables (like x or y), and the operations of addition, subtraction, and multiplication.
Important points to remember about polynomials:

• The degree of a polynomial is the highest exponent of the variable.
• The coefficients are the numbers in front of the variables.
• The constants are terms without variables.

In our example, \(y^{2} - 6y + 8\), the highest exponent is 2, so it's a polynomial of degree 2, also known as a quadratic polynomial. The coefficients are 1 (for \(y^2\)), -6 (for \(y\)), and 8 is the constant term.

A quadratic expression is a specific type of polynomial where the highest exponent is 2. The general form is \(ax^2 + bx + c\). Here, 'a', 'b', and 'c' are constants, and \(x\) is the variable.
In our exercise: \(y^{2} - 6y + 8\), we can see that:

• \(a = 1\)
• \(b = -6\)
• \(c = 8\)

To solve problems involving quadratic expressions, one common approach is to factor them. Factoring quadratics often means finding two expressions that multiply to give the original quadratic.
This helps us solve equations or simplify algebraic expressions. When factoring, the goal is to rewrite the expression as a product of two binomials. For example, our quadratic \(y^{2} - 6y + 8\) factors into \((y-2)(y-4)\).

Factoring quadratics involves several techniques. Let's break down the steps using our example, \(y^{2} - 6y + 8\):
1. **Identify the Polynomial**: Recognize it's a quadratic expression in the form of \(ax^2 + bx + c\).
2. **Find the Factors**: Look for two numbers that multiply to the constant term (8) and add up to the coefficient of the middle term (-6). Here, these numbers are -2 and -4.
3. **Write the Factors**: Rewrite the quadratic expression by splitting the middle term using -2 and -4: \(y^2 - 2y - 4y + 8\).

• This helps us see how to group terms for factoring.

4. **Group Terms**: Group the terms into pairs: \((y^2 - 2y) + (-4y + 8)\).
5. **Factor Each Group**: Find the greatest common factor (GCF) in each group. For \(y^2 - 2y\), it's \(y\); for \(-4y + 8\), it’s -4. This yields \(y(y - 2) - 4(y - 2)\).
6. **Factor Out the Common Binomial**: Notice the common binomial factor is \(y-2\). Factor it out: \((y-2)(y-4)\).
Factoring helps solve quadratic equations and simplify expressions. If a quadratic cannot be factored, it is called 'prime'. Always check your factors by multiplying them back to ensure you get the original quadratic.