Use the remainder theorem to factorise the following expression 2x3 + 9x2 + 7x - 6
Let P(x) = `2x^3 + 3x^2 - 9x - 10` Show P(2) = 16 + 12 - 18 - 10 P(2) = 0 So (x - 2) is a factor Let us divide P(x) with (x-2), we get `(x- 2)(2x^2 + 7x + 5)` This can be further factored to `(x-2)(2x^2 + 5x + 2x + 5)` ……… (Split 7x into two terms, whose sum is 7x and product is10𝑥2) `(x - 2) (2x^2 + 5x + 2x + 5)` `(x - 2))(x(2x+5)+1(2x+5))` (x - 2)(2x + 5)(x + 1) Question: Yvette F. Algebra 8 months, 3 weeks ago Video Answer: Get the answer to your homework problem. Try Numerade free for 7 days DM Oklahoma State University We don’t have your requested question, but here is a suggested video that might help. Related QuestionUse the Remainder Theorem to find the remainder when the function f(x)=x3+6x2-3x is divided by (x+5) DiscussionYou must be signed in to discuss. Video TranscriptIn this problem, we're given the coefficients of X and X and X and X and X and X and X and X and X and X and X and X and X and X and X and X and X and X and X. We're going to divide this by 5. The remainder Theorem is used. If you put five into the minus five, the result will be what the remainder is, which means I can use the synthetic division to do it. I need my coefficients with this. I have a coefficients of one, x squared is six and X is -3 and the constant is zero Let's bring down the one time -5 is -5. One -5 times 1 -5 is 6 -5. I get -8 -5 times -8 so I get 40 for the answer if I had those together. There's our results from the remainder Theorem. X squared Plus X -8 plus 40 over X plus five is the final result. There is a result Step by step solution :Step 1 :Equation at the end of step 1 :(((2 • (x3)) + 32x2) + 7x) - 6 = 0 Equation at the end of step 2 : ((2x3 + 32x2) + 7x) - 6 = 0
Step 3 :Checking for a perfect cube : 3.1 2x3+9x2+7x-6 is not a perfect cube Trying to factor by pulling out :3.2 Factoring: 2x3+9x2+7x-6 Thoughtfully split the expression at hand into groups, each group having two terms : Group 1: 7x-6 Pull out from each group separately : Group 1: (7x-6) • (1) Bad news !! Factoring by pulling out fails : The groups have no common factor and can not be added up to form a multiplication. Polynomial Roots Calculator : 3.3 Find roots (zeroes) of : F(x) = 2x3+9x2+7x-6 Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient In this case, the Leading Coefficient is 2 and the Trailing Constant is -6. The factor(s) are: of the Leading Coefficient : 1,2
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms In our case this means that Polynomial Long Division : 3.4 Polynomial Long Division
Quotient : x2+5x+6 Remainder: 0 Trying to factor by splitting the middle term3.5 Factoring x2+5x+6 The first term is, x2 its coefficient is 1 . Step-1 : Multiply the coefficient of the first term by the constant 1 • 6 = 6 Step-2 : Find two factors of 6 whose sum equals the coefficient of the middle term, which is 5 .
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 2 and 3 Step-4 : Add up the first 2 terms, pulling out like factors : Equation at the end of step 3 : (x + 3) • (x + 2) • (2x - 1) = 0
Step 4 :Theory - Roots of a product :4.1 A product of several terms equals zero. When a product of two or more terms equals zero, then at least one of the terms must be zero. We shall now solve each term = 0 separately In other words, we are going to solve as many equations as there are terms in the product Any solution of term = 0 solves product = 0 as well. Solving a Single Variable Equation : 4.2 Solve : x+3 = 0 Subtract 3 from both sides of the equation : Solving a Single Variable Equation : 4.3 Solve : x+2 = 0 Subtract 2 from both sides of the equation : Solving a Single Variable Equation : 4.4 Solve : 2x-1 = 0 Add 1 to both sides of the equation : Supplement : Solving Quadratic Equation DirectlySolving x2+5x+6 = 0 directly Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula Parabola, Finding the Vertex : 5.1 Find the Vertex of y = x2+5x+6Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 1 , is positive
(greater than zero). Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions. Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of
time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex. For any parabola,Ax2+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is -2.5000 Plugging into
the parabola formula -2.5000 for x we can calculate the y -coordinate : Parabola, Graphing Vertex and X-Intercepts :Root plot for : y = x2+5x+6 Solve Quadratic Equation by Completing The Square 5.2 Solving x2+5x+6 = 0 by
Completing The Square .Subtract 6 from both side of the equation : Now the clever bit: Take the coefficient of x , which is 5 , divide by two, giving 5/2 , and finally square it giving 25/4 Add 25/4
to both sides of the equation : Adding 25/4 has completed the left hand side into a perfect square : We'll refer to this Equation as Eq. #5.2.1 The Square Root Principle says that When two things are equal, their square roots are equal. Note that the square root of Now, applying the Square Root Principle to Eq. #5.2.1 we get: Subtract 5/2 from both sides to obtain: Since a square root has two values, one positive and the other negative Note that √ 1/4 can be written as Solve Quadratic Equation using the Quadratic Formula 5.3 Solving x2+5x+6 = 0 by the Quadratic Formula .According to the Quadratic Formula, x , the solution for Ax2+Bx+C = 0 ,
where A, B and C are numbers, often called coefficients, is given by : -5
± √ 1 Two real solutions: x =(-5+√1)/2=-2.000 or: x =(-5-√1)/2=-3.000 Three solutions were found :
|