Mastering Polynomial Graphs and Circle Theorems

A: Let's jump right in! We're going to review Polynomials and Circle Theorems, tackling each question and figuring out exactly why each answer is what it is, using the official keys. Sound good?

B: Yes! Can we start with the polynomial questions? Sometimes I get confused about what actually makes a function a polynomial.

A: Absolutely. M1, Question 1: 'Which is a polynomial function?' The correct answer is C: f(x) = √2x³ + 5. Only nonnegative integer exponents are allowed for polynomials. Here, x has an exponent of 3, which fits. The others sneak in negatives or roots—those aren't polynomials.

B: Wait... isn't the square root in '√2x³' a problem? That looks tricky.

A: Good eye. But here, only the coefficient is a root—it's √2 times x³, not the root of x. So the exponent on x is still 3, which is allowed.

B: Ahh, okay! Next, zeros and multiplicity always mess me up.

A: Let’s do M1, Q2: 'Find the zeros of P(x) = (x + 4)³(x - 5)⁶.' The answer is A: −4 has multiplicity 3, 5 has multiplicity 6. The zeros come straight from the factors: set each to zero, and their exponents show the multiplicity.

B: So if it’s (x+a)ⁿ, then x = −a, and its multiplicity is n?

A: Exactly right! And for degree, just add all the exponents in the factored form. That's M1, Q4: Correct answer, C: 11. The degree is 1+2+3+5 = 11.

B: Makes sense. So, the degree tells us the maximum number of zeros and also affects the graph's shape, like end behavior?

A: Correct again. End behavior is determined by the degree and the leading coefficient. M1, Q9: It's the Leading Coefficient Test. So, answer C. Even degree means the ends face the same way, sign decides up or down.

B: And turning points—they’re just one less than the degree?

A: Yes, turning points = degree minus one. So a cubic could have up to two turning points. It's a common error to mix that up. Now, let's pivot to circle basics.

B: Sure! Remind me, what exactly is a tangent versus a secant?

A: A tangent touches a circle at exactly one point—no more. A secant cuts through the circle at two points. Got it?

B: Yeah, that helps! How about angles—what's the difference between central and inscribed?

A: A central angle’s vertex is at the circle’s center, and its measure equals the arc it intercepts. An inscribed angle’s vertex is on the circle, and its measure is half its intercepted arc. For example, M2, Q1: 'Which angle has its vertex on a circle and contains chords?' The answer is C: inscribed angle.

B: So if it’s on the circumference, it can only be inscribed. Got it. What about those arc types—minor, major, semicircle?

A: Minor is less than 180°, semicircle is exactly 180°, major is more than 180°. M2, Q5: If an arc’s endpoints are endpoints of the diameter, it’s a semicircle—so answer D.

B: And the sum of central angles always adds to 360°?

A: Right. That’s M2, Q3: answer C. No matter how you slice it, together they fill up the circle.

B: Great! Could we clarify one more circle theorem—like, the diameter bisecting a chord?

A: Of course. For M3: If a diameter is perpendicular to a chord, it bisects both the chord and its corresponding arc. So, perpendicularity is key. And—never mix up tangent and secant: tangent touches just once, and is always ⟂ to the radius at that point, by the theorem in M4.

B: That definitely clears things up! This walkthrough actually ties the formulas and theorems together.

A: Good! Remember, always restate what kind of object you’re dealing with, apply the core rule, and check your concepts—no shortcuts. That way, both polynomials and circles start to feel less like a memorization marathon and more like a story that fits together.