This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — let's simplify these expressions, kamogelo. a) Simplify (p+3)(p+5). Step 1: Use the FOIL method (First, Outer, Inner, Last) to multiply the binomials. (p+3)(p+5) = p · p + p · 5 + 3 · p + 3 · 5 Step 2: Perform the multiplications. p^2 + 5p + 3p + 15 Step 3: Combine like terms. p^2 + 8p + 15 The simplified expression is p^2 + 8p + 15. b) Simplify (k+7)(k-2). Step 1: Use the FOIL method to multiply the binomials. (k+7)(k-2) = k · k + k · (-2) + 7 · k + 7 · (-2) Step 2: Perform the multiplications. k^2 - 2k + 7k - 14 Step 3: Combine like terms. k^2 + 5k - 14 The simplified expression is k^2 + 5k - 14. c) Simplify (c-4)(c-8). Step 1: Use the FOIL method to multiply the binomials. (c-4)(c-8) = c · c + c · (-8) + (-4) · c + (-4) · (-8) Step 2: Perform the multiplications. c^2 - 8c - 4c + 32 Step 3: Combine like terms. c^2 - 12c + 32 The simplified expression is c^2 - 12c + 32. d) Simplify (f+5)(f-5). Step 1: Recognize this as a difference of squares pattern, (A+B)(A-B) = A^2 - B^2. (f+5)(f-5) = f^2 - 5^2 Step 2: Calculate the square. f^2 - 25 The simplified expression is f^2 - 25. e) Simplify (m-8)(m+7). Step 1: Use the FOIL method to multiply the binomials. (m-8)(m+7) = m · m + m · 7 + (-8) · m + (-8) · 7 Step 2: Perform the multiplications. m^2 + 7m - 8m - 56 Step 3: Combine like terms. m^2 - m - 56 The simplified expression is m^2 - m - 56. f) Simplify (r+6)^2. Step 1: Use the square of a binomial formula, (A+B)^2 = A^2 + 2AB + B^2. (r+6)^2 = r^2 + 2(r)(6) + 6^2 Step 2: Perform the multiplications and squaring. r^2 + 12r + 36 The simplified expression is r^2 + 12r + 36. g) Simplify (r-3)^2. Step 1: Use the square of a binomial formula, (A-B)^2 = A^2 - 2AB + B^2. (r-3)^2 = r^2 - 2(r)(3) + 3^2 Step 2: Perform the multiplications and squaring. r^2 - 6r + 9 The simplified expression is r^2 - 6r + 9. h) Simplify (m-7)^2. Step 1: Use the square of a binomial formula, (A-B)^2 = A^2 - 2AB + B^2. (m-7)^2 = m^2 - 2(m)(7) + 7^2 Step 2: Perform the multiplications and squaring. m^2 - 14m + 49 The simplified expression is m^2 - 14m + 49. i) Simplify 7m(7m^2-4n). Step 1: Distribute 7m to each term inside the parenthesis. 7m(7m^2-4n) = (7m)(7m^2) - (7m)(4n) Step 2: Perform the multiplications. 49m^3 - 28mn The simplified expression is 49m^3 - 28mn. j) Simplify (1)/(2)(8x^2+6)(3x^3). Step 1: Multiply the terms inside the first parenthesis by 3x^3. (1)/(2) [ (8x^2)(3x^3) + (6)(3x^3) ] Step 2: Perform the multiplications inside the bracket. (1)/(2) [ 24x^2+3 + 18x^3 ] (1)/(2) [ 24x^5 + 18x^3 ] Step 3: Distribute (1)/(2) to each term inside the bracket. (1)/(2)(24x^5) + (1)/(2)(18x^3) 12x^5 + 9x^3 The simplified expression is 12x^5 + 9x^3. What's next?