Question

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Answer 1

Answer:

The factors of p(x) is (x - 1), (x + 1), (x + 5) and (x - 5)

Step-by-step explanation:

Here we can only solve it by by trial and error, also, it is long,so it might be a bit confusing, "But don't fear, your Friend Mathwiz is here......"

Let's say p(x) = x⁴ - 29x² + 100

Let x = 1

p(1) = 1⁴ - 29(1)² + 100 = 72

Thus,

x = 1 is not a zero of the polynomial

Let x = 2

We are not considering -1 because, here all the powers are even and wouldn't give a negative value.

It means,

p(-1) = (-1)⁴ - 29(-1)² + 100

= 1 - 29 × 1 + 100

= 1 - 29 + 100 = 72

Thus, if the positive values of x is a zero, then its negative value also must be a zero of the polynomial

Now, coming back, x = 2

p(2) = 2⁴ - 29(2)² + 100

= 16 - 29 × 4 + 100

= 16 - 116 + 100

= 0

Then, x = 2 is a zero of the polynomial

thus, x = 2

(x - 2) should be a factor of p(x)

Similarly,

x = -2 is a zero of the polynomial

thus,

(x + 2) is also a factor

because we said that, negative numbers also give positive values and should be a zero

Let the other factor be equal to g(x)

So, now we got,

x⁴ - 29x² + 100 = (x - 2)(x + 2) × g(x)

Using (a - b)(a + b) = a² - b²

p(x) = (x² - 4) × g(x)

thus,

g(x) = p(x) ÷ (x² - 4)

x² - 25

—————————

x² - 4 | x⁴ - 29x² + 100

- (x⁴ - 4x² )

——————

0 - 25x² + 100

- (-25x² + 100)

———————–

0

Thus,

g(x) = x² - 25

We can use simple algebraic identity used above here,

x² - 25 = (x)² - (5)²

We know that,

a² - b² = (a + b)(a - b)

Similarly,

x² - 5² = (x + 5)(x - 5)

Thus,

x⁴ - 29x² + 100 = (x - 1)(x + 1)(x + 5)(x - 5)

So, the factors of p(x) is (x - 1), (x + 1), (x + 5) and (x - 5)

Hope it helped and you understood it........All the best

Answer 2

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Step-by-step explanation:

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