What this lesson covers
- Every factor (x − a) is a zero at x = a, and a plus sign in the factor means a negative zero
- An even exponent means the graph touches the axis and turns; an odd exponent means it cuts through
- End behavior comes from the leading term only: odd degree, opposite ends; even degree, same ends
- A negative leading coefficient flips both tails, and expanding never enters any of it
Questions worked in the video
- 0:47f(x) = (x + 3)(x − 1)(x − 4). What are the zeros of f?
- 1:52f(x) = (x − 2)²(x + 5). At which zero does the graph touch the axis rather than cross it?
- 2:23f(x) = −(x + 1)(x − 3)². Name the zeros, say which one the graph crosses, and describe both tails.
Chapters
- 0:00Do not expand it
- 0:22One fact, three names
- 0:47Read the zeros off the factors
- 1:21The tails come from one term
- 1:52Touch or cross
- 2:23Your turn
- 3:07Three traps
- 3:32Recap
Lesson transcript
The narration of the video, word for word, under its chapter headings.
0:00Do not expand it
Welcome to SAT Climb. Look at this polynomial. Nobody is asking you to expand it. Written this way, it has already told you where its graph hits the x axis. Three numbers, hiding in plain sight.
0:22One fact, three names
The SAT says the same thing three different ways. Zero of the function. Factor of the polynomial. X intercept of the graph. If x minus a is a factor, then f of a is zero, and the graph crosses or touches at a. One fact, three names. When a question changes the wording, it has not changed the work.
0:47Read the zeros off the factors
Here is the money move. f of x equals x plus 3, times x minus 1, times x minus 4. To find a zero you ask, what makes this factor zero. x minus 1 is zero when x is 1. x minus 4 is zero when x is 4. And x plus 3 is zero when x is negative 3. Watch the sign. Plus 3 in the factor means negative 3 on the graph. The zeros are negative 3, 1, and 4, and you never multiplied anything out.
1:21The tails come from one term
Now the tails. End behavior comes from one term, the leading one. Multiply just the first pieces: x times x times x is x cubed. Degree three, odd, and the coefficient is positive, so the graph comes up from the bottom left and leaves through the top right. Odd degree, opposite ends. Even degree, same ends. Negative coefficient flips both. That is the entire rule, and expanding never enters it.
1:52Touch or cross
One more thing the factors tell you. Look at f of x equals x minus 2, squared, times x plus 5. The zeros are 2 and negative 5, but they behave differently. The exponent on x minus 2 is even, so the graph touches the axis at 2 and turns around. The exponent on x plus 5 is odd, so it cuts straight through at negative 5. Even exponent touches. Odd exponent crosses.
2:23Your turn
Your turn. f of x equals negative, x plus 1, times x minus 3, squared. Name the zeros, say which one the graph crosses, and describe the two tails. Pause here. Zeros first. x plus 1 gives negative 1, and x minus 3 gives 3. The exponent on x minus 3 is even, so the graph touches at 3 and crosses at negative 1. The leading term is negative x cubed. Odd degree, negative coefficient, so it starts high on the left and falls to the right.
3:07Three traps
Three traps. One, the sign. x plus 3 is a zero at negative 3, not positive 3. Two, assuming every zero is a crossing. An even exponent means the graph only touches. Three, expanding the whole polynomial to find the tails, when the leading term already told you. Read the factors, not the product.
3:32Recap
Polynomial zeros, solved. Every factor hands you an x intercept, the exponent tells you touch or cross, and the leading term decides both tails. Start free at satclimb.com.
Read the written version: the Nonlinear functions strategy guide, then try 25 hard Nonlinear functions questions with full explanations. Both are free, no account needed.