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## SAT (Fall 2023)

### Course: SAT (Fall 2023) > Unit 10

Lesson 2: Passport to advanced mathematics- Solving quadratic equations — Basic example
- Solving quadratic equations — Harder example
- Interpreting nonlinear expressions — Basic example
- Interpreting nonlinear expressions — Harder example
- Quadratic and exponential word problems — Basic example
- Quadratic and exponential word problems — Harder example
- Manipulating quadratic and exponential expressions — Basic example
- Manipulating quadratic and exponential expressions — Harder example
- Radicals and rational exponents — Basic example
- Radicals and rational exponents — Harder example
- Radical and rational equations — Basic example
- Radical and rational equations — Harder example
- Operations with rational expressions — Basic example
- Operations with rational expressions — Harder example
- Operations with polynomials — Basic example
- Operations with polynomials — Harder example
- Polynomial factors and graphs — Basic example
- Polynomial factors and graphs — Harder example
- Nonlinear equation graphs — Basic example
- Nonlinear equation graphs — Harder example
- Linear and quadratic systems — Basic example
- Linear and quadratic systems — Harder example
- Structure in expressions — Basic example
- Structure in expressions — Harder example
- Isolating quantities — Basic example
- Isolating quantities — Harder example
- Function notation — Basic example
- Function notation — Harder example

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# Polynomial factors and graphs — Basic example

Watch Sal work through a basic Polynomial factors and graphs problem.

## Want to join the conversation?

- Why is it that the examples in these videos are always so easy but the practice problems are so much harder?(34 votes)
- am i the only one who tried it with qudratic formula

and it still worked!(9 votes)- The quadratic formula always works. Factoring is what is sometimes impossible. Still, Sal took way to long.(8 votes)

- I encountered a problem where I needed to find the minimum x-coordinate of the parabola. I forgot the formula and I used the calculator since it was in the calculator section. I got an x = 0.4999999999. With the formula, it was 0.5. Would the answer be marked wrong on the SAT test if I put .499 and not 0.5(as the result from the formula)?(3 votes)
- I believe that it would be marked wrong. Equivalent forms of the expression are permitted, and decimal approximations of repeating fractions or fractions that aren't repeating but can't be represented exactly with the three digits you get, but none of those apply to this. I guess unless the question has some inequality in it, you should automatically assume that 0.49999 means .5.(6 votes)

- all the topics are not covered so its hard to understand the quizzes(5 votes)
- wouldn't zeroes just generally mean all intercepts?(2 votes)
- x-intercepts = zeros = roots(2 votes)

- What is the definition of a polynomial? I tried looking it up but I don't understand what google gives me.(0 votes)
- A polynomial is a string of terms. These terms each consist of x raised to a whole number power and a coefficient. As an example, take the polynomial 4x^3 + 3x + 9. Since this has three terms, it's called a trinomial. Two-term polynomials are binomials and one-term polynomials are monomials. The 9 term would technically be multiplied to x^0, but since that is 1 we leave it out. We'd call this polynomial a 3rd-degree polynomial because its highest exponent of x is 3. The roots or zeroes of a polynomial are its x-intercepts when it's plotted on a graph.

The two main rules of polynomials are that you can only have one variable, and that only whole number powers are allowed. Polynomials don't have negative powers, because then you would have a variable in the denominator. Fractional powers also don't count.

We can use polynomials to model real life situations, and they all have predictable graphs. Odd-degree polynomials will start from the top/bottom of the graph on the -x side and end up at the opposite side on the +x side of the graph. Even-degree polynomials will look like a "u": As x goes towards negative infinity, the y value will either be extremely positive or negative, and it will be on the same side for when x tends towards positive infinity.

Hopefully this is a good enough starting point.(4 votes)

- Would you need to factor the problem out, it seemed like a trick question?(1 vote)
- No you dont have to cause it already is, this is just an easy example.(1 vote)

- wont we check the number of turning/stationary points? What type of question will we be asked in order to check that?(1 vote)
- do you mean when the derivative is 0( peaks where the graph does like a valley or a hill)(1 vote)

- I didn't understand how to solve this. I'm getting incorrect answers :((0 votes)
- how to solve these problems when graph is not givin?(0 votes)
- The graph is your answer choice here. If you're asked to identify a polynomial with 4 distinct zeroes and given an equation for example, the correct answer would be able to be split into 4 different factors that look like (x-r), with obviously different numbers for the r-value.

The root form of a polynomial, y = a(x - r1)(x - r1)... gives you all the roots like that, in factored form.(0 votes)

## Video transcript

- [Instructor] The polynomial p of x has four distinct zeros,
four distinct zeros. Which of the following graphs could represent y is equal to p of x. Well, four distinct zeros
means that the graph is going to intersect the
x-axis exactly four times. So this first one intersects the x-axis one, two, three times. So we can rule that one out. This one intersects the x-axis one, two, three, four, five times. Let's go and rule that one out. This one intersects the x-axis one, two, three, four times. It has four distinct zeros. So this looks like,
this one right over here looks like our choice, and
this one right over here, it actually only has two distinct zeros, or at least that's what it looks like. So I'd rule that one out as well. So this is definitely going to be the one that has four distinct zeros. Zeros are the x values
at which the function is equal or the polynomial, in this case, which is a function of x, is
going to be equal to zero, and you can see that's
happening exactly four times.