Polynomial factorization #1 - Questions and Answers

Question 1 Write the polynomial as the product of linear factors and list all the zeros of the function. g(x) = x³-3x²+x+5 Answer 1 Step 1 Given: The polynomial function g(x) = x³-3x²+x+5, To determine: The factorization and zeros of polynomial Step 2 Explanation: Factorizing the given polynomial function: Here it can be observed that g(-1) = (−1)³ – 3(−1)² + (-1) + 5 ⇒ g(-1) = −1-3-1+5= 0 Therefore (x+1) is a factor of the polynomial 3 - 3x2 + x + 5, Therefore polynomial x³-3x² + x + 5 can be written as x³-3x2 + x + 5 = x²(x + 1) - 4x(x+1)+5(x+1) ⇒ x³-3x2 + x + 5 = (x + 1)(x² - 4x + 5) Here x2 - 4x + 5 is irreducible. Therefore given polynomial function can be factorized as g(x) = (x + 1)(x² - 4x + 5) Step 3 Finding the zeros of the given polynomial function: Solving g(x)=0 for x, g(x)=0 ⇒ (x + 1)(x2 - 4x + 5) = 0 ⇒ x = -1 [here x2 - 4x + 5≠0] Therefore x=-1 is the only zero of the given polynomial function. Answer 2 We can start by using the Rational Root Theorem to identify potential rational zeros. According to the theorem, the possible rational zeros of g(x) are of the form ±p/q, where p is a factor of the constant term (in this case, 5) and q is a factor of the leading coefficient (in this case, 1). The factors of 5 are +1 and ±5, while the factors of 1 are ±1. Therefore, the possible rational zeros of g(x) are ±1 and ±5. Next, we can use synthetic division or long division to check these potential zeros. Let's start with x = 1: 1 | 1 -3 1 5 | 1 -2 -1 ---------------- Since the remainder is not zero, x = 1 is not a zero of g(x). Let's try x = -1: -1 | 1 -3 1 5 | -1 4 -5 ---------------- 1 -4 5 0 Again, the remainder is not zero, so x = -1 is not a zero either. Let's continue with x = 5: 5 | 1 -3 1 5 | 5 10 55 ---------------- 1 2 11 60 Once again, the remainder is not zero, meaning x = 5 is not a zero. Finally, let's check x = -5: -5 | 1 -3 1 5 | -5 40 -205 ----------------- 1 -8 41 -200 The remainder is zero, which indicates that x = - 5 is a zero of g(x). Now, we can factor g(x) using the zero we found: g(x) = (x-(-5)) (x² + 5x +39) The quadratic factor x² + 5x + 39 does not have real roots, so we cannot further factor it using linear factors. Therefore, the polynomial g(x) expressed as a product of linear factors is: g(x) = (x + 5)(x² + 5x + 39) The zeros of g(x) are x = -5 (with multiplicity 1) and the complex zeros of the quadratic factor x² + 5x + 39. Answer 3 Step 1: To write the polynomial as the product of linear factors and list all the zeros of the function g(x) = x³-3x² + x + 5, we can use the factor theorem and synthetic division. First, we find the possible rational zeros by considering the factors of the constant term (5) divided by the factors of the leading coefficient (1). The possible rational zeros are ±1 and ±5. Using synthetic division, we test these possible zeros. By substituting each value into the polynomial, we find that x = -1 is a zero. Performing synthetic division with x = -1, we get: -1 | 1 -3 1 5 | -1 4 -5 ----------------- 1 -4 5 0 Step 2: The result shows that x = -1 is a zero of g(x), and the resulting quotient is x² - x + 5. We can now write the polynomial as the product of linear factors: g(x) = (x + 1)(x² - x + 5) To find the remaining zeros, we solve the quadratic factor x2 - x + 5 by factoring or using the quadratic formula. However, in this case, the quadratic factor has no real roots since the discriminant (b² - 4ac) is negative. Therefore, the zeros of g(x) are x = -1 and two complex roots from the quadratic factor. Question 2 What is y=2x-4 in standard-form? Answer 1 The answer is -2x+y=-4 or 2x-y=4. Standard form is Ax+Ay=C. Convert y=2x-4 to standard form. Subtract 2x from both sides. y-2x=-4 Place the x term in front of the y term. -2x+y=-4 You could also multiply the equation times -1 (optional). 2x-y=4 Answer 2 Solution -2x+y=-4 or 2x-y=4 Standard form is Ax+By=C Convert y=2x-4 to standard form. Subtract 2x from both sides. y-2x=-4 Place the x term in front of the y term. -2x+y=-4 You could also multiply the equation times -1 2x-y=4 Question 3 When b=3 and a=6, a linear regression equation results. What will happen to Y when X equals 4? Answer Step 1 We'll locate the solution after first identifying the regression equation. Step 2 y = a + (b) x given, a=-6, b=3 So, y = - 6 + 3x For x=4 y = -6+3*4= -6+12=6 Hence, the predicted value is ŷ = 6 Question 4 Find the prime factorization of 56. Answer Step 1 Consider the number 56. Since 56 is an even number, it is divisible by the prime number 2. Note that, 56 = 2 × 28. We proceed further by factorising composite factors after each step. Next, we have to consider the factorisation of the composite factor 28 of 56. Since 28 is an even number, it is divisible by the prime number 2. Also, 28 = 2 × 14. Substituting 28 = 2 × 14 ∈ 56 = 2 × 28, we get 56 = 2 × 2 × 14. Step 2 Next, we have to consider the factorisation of the composite factor 14 of 56. Since 14 is an even number, it is divisible by the prime number 2. Also, 14 = 2 x 7. Substituting 14 = 2 × 7 ∈ 56 = 2 × 2 × 14, we get 56 =2×2×2×7. Note that, all the factors in the right hand side of 56 =2×2×2×7 are primes. Hence, we stop here. Note that, 56 = 2 × 2 × 2 × 7 = 2³ x 7. Thus, 56 = 2 x 28 = 2 × 2 × 14 =2×2×2×7 The prime factorisation of 56 is 2 x 2 x 2×7 or 2³ x 7. Question 5 How do you write y=3x-8 in standard form? Answer The answer is 3x-y=8. Standard form for a linear equation is Ax+By=C. Write y=3x-8 in standard form. Subtract 3x from both sides of the equation. -3x+y=-8 Multiply both sides by -1. 3x-y=8