Polynomials

Question 1

How do you find the subject of the formula in a given equation?

Answer 1

1. Identify the subject: Determine which variable you want to solve for.
2. Isolate the subject: Use inverse operations to get the subject by itself on one side of the equation.
3. Simplify: Perform any necessary simplifications to express the subject clearly.

Question 2

How do you apply the factor and remainder theorems to factorize a given expression?

Answer 2

1. Remainder Theorem: Substitute 𝑥 = 𝑎 into the polynomial 𝑓(𝑥) to find 𝑓(𝑎). If
   𝑓(𝑎)=0, then 𝑥−𝑎 is a factor.
2. Factor Theorem: Use the factor 𝑥 − 𝑎 to divide the polynomial and find the quotient.
3. Factorize: Write the polynomial as the product of 𝑥 − 𝑎 and the quotient.

Question 2

How do you multiply and divide polynomials of degree not more than 3?

Answer 2

1. Multiplication:
   * Distribute each term of the first polynomial to each term of the second polynomial.
   * Combine like terms.
2. Division:
   * Divide the leading term of the dividend by the leading term of the divisor.
   * Multiply the entire divisor by this quotient and subtract from the dividend.
   * Repeat until the degree of the remainder is less than the degree of the divisor.

Question 3

How do you factorize by regrouping difference of squares, perfect squares, and cubic expressions?

Answer 3

1. Difference of Squares: Use 𝑎² − 𝑏² = (𝑎 − 𝑏) (𝑎 + 𝑏).
   Perfect Squares: Identify patterns like 𝑎² + 2𝑎𝑏 + 𝑏² = (𝑎 + 𝑏)².
2. Cubic Expressions: Use identities like
   𝑎³ − 𝑏³ = (𝑎 − 𝑏)(𝑎² + 𝑎𝑏 + 𝑏²).

Question 4

How do you solve simultaneous equations involving one linear and one quadratic equation?

Answer 4

1. Substitute: Solve the linear equation for one variable and substitute into the quadratic equation.
2. Simplify: Simplify the quadratic equation and solve for the remaining variable.
3. Back-substitute: Use the found value to solve for the other variable in the linear equation.

Question 5

How do you interpret graphs of polynomials of degree not greater than 3?

Answer 5

1. Plot Points: Calculate and plot points using different values of 𝑥.
2. Identify Key Features: Look for intercepts, turning points, and end behavior.
3. Analyze: Use the graph to identify maximum and minimum values and understand the behavior of the polynomial.