Which of the following is a zero of the polynomial 3x² 8x 5 * 0?
Solution : `f(x)=(3x^2-8x+5)(ax+b)+(cx+d)` Show
Step by step solution :Step 1 :Equation at the end of step 1 :(3x2 - 8x) + 5 = 0Step 2 :Trying to factor by splitting the middle term2.1 Factoring 3x2-8x+5 The first term is, 3x2 its coefficient is
3 . Step-1 : Multiply the coefficient of the first term by the constant 3 • 5 = 15 Step-2 : Find two factors of 15 whose sum equals the coefficient of the middle term, which is -8 .
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -5 and -3 Step-4 : Add up the first 2 terms, pulling out like factors : Equation at the end of step 2 :(3x - 5) • (x - 1) = 0Step 3 :Theory - Roots of a product :3.1 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 : 3.2 Solve : 3x-5 = 0 Add 5 to both sides of the equation : Solving a Single Variable Equation : 3.3 Solve : x-1 = 0 Add 1 to both sides of the
equation : Supplement : Solving Quadratic Equation DirectlySolving 3x2-8x+5 = 0 directlyEarlier 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 : 4.1 Find the Vertex of y = 3x2-8x+5Parabolas 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, 3 , 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 1.3333 Plugging into
the parabola formula 1.3333 for x we can calculate the y -coordinate : Parabola, Graphing Vertex and X-Intercepts :Root plot for : y = 3x2-8x+5 Solve Quadratic Equation by Completing The Square 4.2 Solving 3x2-8x+5 = 0 by Completing The
Square .Divide both sides of the equation by 3 to have 1 as the coefficient of the first term : Subtract 5/3 from both side of the equation : Now the clever bit: Take the coefficient of x , which is 8/3 , divide by two, giving 4/3 , and finally square it giving 16/9 Add 16/9 to both sides of the equation : Adding 16/9 has completed the left hand side into a perfect square : We'll refer to this Equation as Eq. #4.2.1 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. #4.2.1 we get: Add 4/3 to both sides to obtain: Since a square root has two values, one positive and the other negative Note that √ 1/9 can be written as Solve Quadratic Equation using the Quadratic Formula 4.3 Solving 3x2-8x+5 = 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 : 8
± √ 4 Yes! The prime factorization of 4 is √ 4 = √ 2•2 = So now we are looking at: Two real solutions: x =(8+√4)/6=(4+)/3= 1.667 or: x =(8-√4)/6=(4-)/3= 1.000 Two solutions were found :
Which of the following is a zero of the polynomial 3x2 5?Thus, the zeroes of the given polynomial 3x2 + 5x – 2 are – 2 and 1/3.
What is the factorization of 3x2 8x 5?Summary: The factorization of 3x2 - 8x + 5 is (3x - 5)(x - 1).
Which of the following is a zero of the polynomial?Hence, zero polynomial can be written as p(x)=0.
How do you find zeros of a polynomial?The different types of equations and the methods to find their zeros of polynomial are as follows. Linear Equation: A linear equation is of the form y = ax + b. The zero of this equation can be calculated by substituting y = 0, and on simplification we have ax + b = 0, or x = -b/a.
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