Reading over the list of topics included on the math section of the new SAT can seem like slogging through a knee-deep vat of jargon. Buried in that list is the phrase “understand the relationships between zeros and factors of polynomials.” Although it sounds like the sort of thing writers on a sci-fi show dump in to make a scene seem technical, (“Sorry, Captain, the relationship between the zeros and the polynomial factors in the engine is too much, we just don’t have the power…”), it’s actually a relatively simple concept that should be familiar from your high school math classes.
What is a “Zero”
Polynomials are mathematical expressions that contain multiple algebraic terms and generally include multiple terms where the same variable is raised to different powers. Graphing a polynomial can tell us many things about the expression itself, including whether it has any “zeros.” Zeros occur anytime the graph of a polynomial crosses the x-axis, because at those points y has a value of 0. Sometimes these “zeros” are also referred to as “roots.”
How many zeros?
According to the Fundamental Theorem of Algebra, the number of zeros a polynomial will have is equal to the degree of the polynomial. While this can be a little misleading (some of those roots may be “imaginary”), it can still help us set expectations for how we expect the graph of a polynomial to behave–our polynomial will have at most the same number of “real” roots as its degree.
For example, consider a first-order polynomial where x is only raised to the first power (remember that x^1 = x). Our polynomial would have the form y = ax + b. When a first-order polynomial is graphed, it forms a straight line where the slope is the coefficient of the variable x (a) and the y intercept is the constant at the end of the equation (b). Of course, a straight line can only cross the x axis once so we know that first order equations must have only a single root.
Now consider a second-order polynomial, also known as a quadratic equation, where x^2 and x^1 terms both appear. We can write out a generic quadratic equation as y = ax^2 + bx + c. When a quadratic equation is graphed, it forms a parabola. Sometimes the tip of a quadratic parabola is above the x axis with both ends pointing up (or below the x axis with both ends pointing down)–it never crosses the x axis. Sometimes the tip of a quadratic parabola may be located directly on the x axis, intersecting the axis exactly once and never again. And sometimes the tip of a quadratic parabola is located above or below the x axis with both its ends pointing towards the axis, crossing the axis twice. At the very most, a quadratic equation can have two roots.
Factoring to Find Zeros
The SAT expects that you can sketch the graph of a simple polynomial given certain provided information. Being able to identify where x axis intercepts occur is an important part of being able to visualize the graph of a polynomial and factoring can be an extremely useful tool for finding those zeros.
If we can break a polynomial expression up into factors, then we know that if any of those factors are set to zero the entire expression will equal zero and we’ve found one of our roots. Consider this example from a first degree polynomial:
y = ax + b – initial polynomial
y = a(x + b/a) – factored polynomial
Notice that if we distribute the a to each of the terms inside the parenthesis we get the original polynomial. If we want y to equal 0, then we need the portion of the equation inside the parenthesis to also equal 0, which happens if x = -b/a. We now know that -b/a must be where the line crosses the x axis.
Now let’s look at a quadratic equation, y = ax^2 + bx + c. Once again, our roots will occur where y is equal to zero. As mentioned earlier, at the very most there will be two “real” zeros to find. On most SAT problems, ultimately quadratic equations factor into two terms, each containing a constant plus or minus a constant times x.
For example:
y = -8x^2 – 33x + 12 → y = (3 – 9x)(4 + x)
y = 15x^2 +55x + 30 → y = (2 + 3x)(15 + 5x)
Once a quadratic equation is factored into two terms, adjusting x so that either term equals zero will also make y zero and is a root of the equation. Using the same examples as above:
y = -8x^2 – 33x + 12 → y = (3 – 9x)(4 + x) Zeros: x = ⅓ and x = -4
y = 15x^2 +55x + 30 → y = (2 + 3x)(15 + 5x) Zeros: x = ⅔ and x = -3
Based on this information you would be able to identify where the parabola crossed the x axis.
Factoring is the reverse action of FOILing and thinking through that process may be helpful when attempting to factor an equation. (Click here to read a post on FOILing). When attempting to factor a quadratic equation, it may be helpful to remember that if the constant in the original equation is positive, the two terms in parenthesis will either both involve addition or both involve subtraction. If the constant is negative, then one of the terms will involve addition, while the other one will involve subtraction.
Conclusion
The new SAT math section intentionally pushes test-takers to use higher level math skills, including being able to interpret information about graphs based on their equation. Want more practice finding zeros? Talk to a Curvebreakers tutor to gain access to test preparation materials specifically designed to make sure you don’t get caught off guard by the new test.