What is a depressed polynomial?
When a polynomial is divided by one of its binomial factors x – r, the quotient is called a depressed polynomial. A depressed polynomial has a degree less than the original polynomial. In Example 3, x – 1 is a factor of 2×3 – 3×2 + x. Use synthetic division to find the depressed polynomial.
What does it mean to depress a function?
When you divide a polynomial by one of its binomial factors, the quotient is called a depressed polynomial. Here the quotient or the depressed polynomial is x2+3x−10 . From the results of the division and by using the Remainder Theorem , we can write the following statement.
What is depressed cubic equation?
Depressing the Cubic Equation To depress the cubic equation, we substitute. x = y − b 3 a . x=y-\frac{b}{3a}.
What is the difference between factor and Remainder Theorem?
Basically, the remainder theorem links remainder of division by a binomial with the value of a function at a point, while the factor theorem links the factors of a polynomial to its zeros.
What is the discriminant of a cubic equation?
Δ = b² – 4ac. If the discriminant Δ is zero, the equation has a double root, i.e. there is a unique x that makes the equation zero, and it counts twice as a root. If the discriminant is not zero, there are two distinct roots.
How do you solve depression cubes?
Solving the depressed cubic. (s-t)3+3st (s-t)=s3-t3. This is true since we can simplify the left side by using the binomial formula to: (s3-3s2t+3st2-t3)+(3s2t-3st2)=s3-t3. From this, we can find a value for u by the quadratic formula, then obtain t, afterwards s and we’re done.
What is the remainder theorem equation?
The remainder theorem formula is: p(x) = (x-c)·q(x) + r(x). The basic formula to check the division is: Dividend = (Divisor × Quotient) + Remainder.
What is the meaning of remainder theorem?
Definition of remainder theorem : a theorem in algebra: if f(x) is a polynomial in x then the remainder on dividing f(x) by x − a is f(a)
What is a depressed cubic?
An important note in the solution of the cubic (meaning developing a formula) is the shift from a general cubic to a form that admitted a solution. This special form is called the depressed cubic, where the coefficient of the quadratic term is zero.
Why is discriminant called discriminant?
i.e., it discriminates the solutions of the equation (as equal and unequal; real and nonreal) and hence the name “discriminant”. It is usually denoted by Δ or D. The value of the discriminant can be any real number (i.e., either positive, negative, or 0).
What is depression cubic?
What is remainder theorem explain with example?
For example: if f(a) = a3-12a2-42 is divided by (a-3) then the quotient will be a2-9a-27 and the remainder is -123. if we put, a-3 = 0. then a = 3. Hence, f(a) = f(3) = -123. Thus, it satisfies the remainder theorem.
What is difference between remainder theorem and factor theorem?
What is remainder theorem class 10th?
Remainder Theorem is an approach of Euclidean division of polynomials. According to this theorem, if we divide a polynomial P(x) by a factor ( x – a); that isn’t essentially an element of the polynomial; you will find a smaller polynomial along with a remainder.
What is difference between factor theorem and remainder theorem?
What is the discriminant equation?
The discriminant is the part of the quadratic formula underneath the square root symbol: b²-4ac. The discriminant tells us whether there are two solutions, one solution, or no solutions.