Polynomial factorization #19 - Questions and Answers

Question 1 Give a polynomial division that has a quotient of x+5 and a remainder of -2. Answer Step 1 The quotient is x+5 and the remainder is -2 Using the equation, Polynomial = Divisor × Quotient + Remainder Take divisor = x + 1 Consequently, the polynomial is Polynomial = (x + 1)(x + 5) + (-2) Polynomial = x² + 6x + 5 - 2 Polynomial = x² + 6x + 3 Step 2 With the aid of polynomial division, x² + 6x + 3 x + 1 The first term is x² so the first term in the quotient is x x(x + 1) = x² + x So, x² + 6x + 3 - (x² + x) = 5x + 3 Now if 5x + 3 divide by x + 1 then the quotient is 5 5(x+1) = 5x + 5 So, 5x + 3 - (5x + 5) = -2 (This is the remainder) The quotient is x+5 and remainder is -2 Hence, the solution Question 2 How do you determine the degree of a polynomial? Answer 1 Explanation: The degree of a polynomial is determined by the highest power of x in the polynomial f(x) = 2x⁵ - 3x² + 4 The highest power of x in f(x) is 5. Hence, degree of f(x)=5 Apolynomial with only a constant, e.g. P(x)=5 has a degree of 0 as 5 = 5x⁰ Answer 2 Explanation: ab² + a²b³ = a³b¹ To find the degree of the polynomial, you first have to identify each term [term is for example ab²], so to find the degree of each term you add the exponents. EX: ab² = a¹b² = 1+2=3-Degree of 3 Highest degree is 5 → a²b³ = 2+3=5 Question 3 How do you simplify polynomials? Answer 1 You can simplify polynomials only if they have roots. You can think of polynomials as numbers, and of monomials of the form (x-a) as 'numbers. So, as you can write a composite numbers as product of 's, you can write a "composite" polynomial as product of monomials of the form (x-a), where a is a root of the polynomial. If the polynomial has no roots, it means that, in a certain sense, it is "", and cannot thus be further simplified. For example, x² + 1 has no (real) roots, so it cannot be simplified. On the other hand, x² - 1 has roots ±1, so it can be simplified into (x+1)(x-1). Finally, x³ + x has a root for x=0. So, we can write as x(x² + 1), and for what we saw before, this expression is no longer simplifiable. Answer 2 To simplify a polynomial, we have to do two things: 1) combine like terms, and 2) rearrange the terms so that they're written in descending order of exponent. First, we combine like terms, which requires us to identify the terms that can be added or subtracted from each other. Question 4 What is the difference between a monomial, binomial and polynomial? Answer 1 You can find the answer in the name. A monomial is An algebraic equation where in its simplified form has only one term A binomial is An algebraic equation where in its simplified form has only 2 terms A Trinomial is An algebraic equation where in its simplified form has only 3 A polynomial is An algebraic equation where in its simplified form has n number of terms. Answer 2 A monomial has no variables in its denominator and will only have one term. A binomial is the sum of two monomials and thus will have two unlike terms. A trinomial is the sum of three monomials, meaning it will be the sum of three unlike terms. A polynomial is the sum of one or more terms. Question 5 What is the difference between standard form, vertex form, factored form? Answer 1 Standard form: y = ax² + bx + c for some constants a,b,c Vertex form: y = m(x − a)² + b for some constants m,a,b (the vertex is at (a,b)) Factored form: y=(ax+b)(cx+d) or possibly y=m(ax+b)(cx+d) for some constants a,b,c,d (and m) Answer 2 Standard form is ax² + bx + c. Vertex form is a(x – h)² + k, which reveals the vertex and axis of symmetry. Factored form is a(x-r)(x-s), which reveals the roots.