Introduction to polynomials · Learner pages

Lesson 1 · Read the structure of an expression

Identify terms, coefficients and the degree of a polynomial.

Name or initials: ____________   Date: ____________

A variable stands for a number; a coefficient multiplies a variable.

1. Separate signed terms. In 4x² − 3x + 8, the terms are 4x², −3x and 8. Keep the minus sign with −3x.
2. Read coefficients. The coefficients of x² and x are 4 and −3. The constant term is 8.
3. Find the degree. The largest exponent is 2, so this is a quadratic polynomial. The x term has exponent 1.
4. Test what is allowed. Polynomial exponents are nonnegative integers. Thus 2/x = 2x⁻¹ is not a polynomial in x. A nonzero constant has degree 0; the zero polynomial has no degree assigned here.

Worked example: For 7 − 2t + 5t³, the coefficients are 7, −2 and 5. Rearranging gives 5t³ − 2t + 7. Its degree is 3, not the number of written terms.

1. What is the degree of 6x³ + x − 2?

   1. 3 · 2. 6 · 3. 2

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

   1. 4 · 2. 7 · 3. −7

3. Which is a polynomial in x?

   1. 1/x + 2 · 2. √x + 1 · 3. 3x² − 5

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

Lesson 2 · Evaluate and check a value

Substitute a value consistently and respect the order of operations.

Name or initials: ____________   Date: ____________

Square before multiplying a squared term by its coefficient.

1. Write the rule. Let p(x) = 2x² − 5x + 2. To evaluate p(−1), replace every x with −1.
2. Use parentheses. p(−1) = 2(−1)² − 5(−1) + 2. Parentheses show that the whole negative number is squared.
3. Calculate in order. (−1)² = 1, so the result is 2 + 5 + 2 = 9.
4. Check a proposed zero. For x = 2, p(2) = 2×4 − 10 + 2 = 0. Therefore 2 is a zero. This does not mean p is zero for every input.

Worked example: For q(t) = 3t − 12, q(5) = 15 − 12 = 3. Solving 3t − 12 = 0 gives t = 4, and q(4) = 0 confirms it.

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

   1. 11 · 2. 14 · 3. 9

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

   1. 0 · 2. 3 · 3. 9

3. For p(x)=x²+x, p(−2) equals…

   1. −6 · 2. 2 · 3. 6

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

Lesson 3 · Build a linear model

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

Name or initials: ____________   Date: ____________

A table can show how an output changes when the input increases by one.

1. Describe the situation. A craft kit costs ₹20 to prepare plus ₹6 for each card. For n cards, the total is C(n)=20+6n.
2. Build a table. For n=0,1,2,3 the totals are 20,26,32,38 rupees. Each extra card adds exactly ₹6.
3. Interpret a graph. Points (0,20),(1,26),(2,32) lie on the line C=6n+20. Here actual orders use nonnegative whole-number n.
4. Test the model. For five cards C(5)=20+30=₹50. The rule assumes the preparation charge and per-card cost stay fixed; bulk discounts would require a different rule.

Worked example: A tank begins with 90 L and loses 5 L each minute at a steady rate. V(t)=90−5t. After 8 minutes it contains 50 L. This model applies only while the tank is draining, 0≤t≤18.

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

   1. Cost per card · 2. Starting charge · 3. Number of cards

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

   1. 44 · 2. 104 · 3. 30

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

   1. Increase 90 L · 2. Decrease 5 L · 3. Decrease 90 L

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

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?

   8 / 4 / 1

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

   2 / x² / 9

3. For q(t)=4t−20, which is a zero?

   4 / 20 / 5

4. If p(x)=3x², p(−2) equals…

   −12 / 12 / 36

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

   y=8x+3 / y=3x+8 / y=11x

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

   Water volume cannot become negative / All lines stop at 18 / The rate changes at 8

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 / x² + 1 / x³

2. Why is x⁻² not a polynomial in x?

   Its coefficient is 1 / Its exponent is negative / It has one term

3. For q(t)=5−t, q(7) equals…

   2 / 12 / −2

4. A proposed zero r must satisfy…

   p(r)=0 / r=0 always / p(0)=r

5. For C(n)=12+4n, C(3) equals…

   24 / 48 / 19

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

   2,4,8 / 3,8,13 / 1,3,6