Introduction to polynomials · Answer guide

Lesson 1 · Read the structure of an expression

Identify terms, coefficients and the degree of a polynomial.

1. What is the degree of 6x³ + x − 2?

   Answer: 3. The highest variable exponent is 3.

2. What is the coefficient of x in 4x² − 7x + 1?

   Answer: −7. The minus sign belongs to the x term, so its coefficient is −7.

3. Which is a polynomial in x?

   Answer: 3x² − 5. 3x² − 5 has only nonnegative integer powers of x.

Discuss: Write two different degree-2 polynomials and explain what makes their degrees equal.

Teaching note: A negative coefficient is allowed; a negative power of the variable is not. Open responses need discussion; they are not automatically marked.

Lesson 2 · Evaluate and check a value

Substitute a value consistently and respect the order of operations.

1. If p(x)=2x+3, find p(4).

   Answer: 11. 2×4+3=11.

2. If p(x)=x²−9, which input is a zero?

   Answer: 3. 3²−9=0. The input −3 is another zero.

3. For p(x)=x²+x, p(−2) equals…

   Answer: 2. (−2)²+(−2)=4−2=2.

Discuss: Invent a linear polynomial with zero 6, then test another input.

Teaching note: −1² means −(1²), but (−1)² means (−1)×(−1). Parentheses matter. Open responses need discussion; they are not automatically marked.

Lesson 3 · Build a linear model

Explain a constant rate and a starting value using a linear expression.

1. For C(n)=20+6n, what does 20 represent?

   Answer: Starting charge. C(0)=20, the charge before any cards are added.

2. Find C(4) when C(n)=20+6n.

   Answer: 44. 20+6×4=44.

3. For V(t)=90−5t, what is the change each minute?

   Answer: Decrease 5 L. The coefficient −5 describes a loss of 5 L per minute.

Discuss: Compare a ₹10 setup charge plus ₹8 per item with a ₹30 setup charge plus ₹4 per item. When do the totals match?

Teaching note: The starting fee is added once. It is not multiplied by the number of cards. Open responses need discussion; they are not automatically marked.

Final check · Set 1

Try one set after the lessons. Use the other set after reviewing any errors.

1. What is the degree of 8t⁴ − t?

   4. The highest exponent with nonzero coefficient is 4.

2. What is the constant term of 2x² + 9?

   9. The term independent of x is 9.

3. For q(t)=4t−20, which is a zero?

   5. 4×5−20=0.

4. If p(x)=3x², p(−2) equals…

   12. 3×(−2)²=3×4=12.

5. A rule gives outputs 8,11,14 for inputs 0,1,2. Which rule fits?

   y=3x+8. The starting output is 8 and each step adds 3.

6. Why limit V(t)=90−5t to at most 18 minutes here?

   Water volume cannot become negative. The tank is empty at t=18. Extending the rule would predict impossible negative volume.

Final check · Set 2

Try one set after the lessons. Use the other set after reviewing any errors.

1. Which expression is linear in x?

   6x − 1. 6x − 1 has degree 1.

2. Why is x⁻² not a polynomial in x?

   Its exponent is negative. A polynomial in x uses only nonnegative integer exponents.

3. For q(t)=5−t, q(7) equals…

   −2. 5−7=−2.

4. A proposed zero r must satisfy…

   p(r)=0. A zero is an input whose output is 0.

5. For C(n)=12+4n, C(3) equals…

   24. 12+4×3=24.

6. Which table shows a constant rate for inputs 0,1,2?

   3,8,13. Both output increases are 5.